diff --git a/papers/project_paper_1_relativity/references/Bombelli2009.pdf b/papers/project_paper_1_relativity/references/Bombelli2009.pdf new file mode 100644 index 00000000..28be1f86 --- /dev/null +++ b/papers/project_paper_1_relativity/references/Bombelli2009.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:7314322f3517e9523e98a59a35a263db650a0659edd0df1d174e084606884119 +size 461293 diff --git a/papers/project_paper_1_relativity/references/Bombelli2009.txt b/papers/project_paper_1_relativity/references/Bombelli2009.txt new file mode 100644 index 00000000..1d7a654e --- /dev/null +++ b/papers/project_paper_1_relativity/references/Bombelli2009.txt @@ -0,0 +1,3668 @@ +arXiv:0810.0096v3 [math.OA] 20 Feb 2012 + +C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: +FILTRATED K-THEORY + +RALF MEYER AND RYSZARD NEST + +Abstract. We define the filtrated K-theory of a C∗-algebra over a finite topo- +logical space X and explain how to construct a spectral sequence that computes +the bivariant Kasparov theory over X in terms of filtrated K-theory. +For finite spaces with totally ordered lattice of open subsets, this spectral +sequence becomes an exact sequence as in the Universal Coefficient Theorem, +with the same consequences for classification. +We also exhibit an example where filtrated K-theory is not yet a complete +invariant. We describe two C∗-algebras over a space X with four points that +have isomorphic filtrated K-theory without being KK(X)-equivalent. For this +space X, we enrich filtrated K-theory by another K-theory functor to a com- +plete invariant up to KK(X)-equivalence that satisfies a Universal Coefficient +Theorem. + +1. Introduction + +1.1. The UCT-problem. One of the main problems in the theory of C∗-algebras +is the computation of the equivariant KK-theory of C∗-algebras endowed with some +extra structure. Here we apply the general techniques developed in [6,9] to the case +of C∗-algebras with a non-trivial ideal lattice. The appropriate version of KK-theory +is Kirchberg’s generalisation of Kasparov theory to C∗-algebras over non-Hausdorff +topological spaces (see [5]). Our goal is to compute it in terms of more manageable +K-theoretic information, generalising the usual Universal Coefficient Theorem that +computes Kasparov’s original theory for C∗-algebras in the bootstrap class by an +exact sequence + +(1.1) +Ext +� +K∗+1(A), K∗(B) +� +֌ KK∗(A, B) ։ Hom +� +K∗(A), K∗(B) +� +. + +The generalisation of the bootstrap class to the case of C∗-algebras with non- +trivial ideal lattice was introduced and studied in [8]. Let us first recall some of +the notation from [8]. Let X be a (usually non-Hausdorff) topological space. A +C∗-algebra over X is a C∗-algebra A endowed with a continuous map Prim(A) → X. +Let C∗alg(X) be the category of C∗-algebras over X; the morphisms in C∗alg(X) are +given by X-equivariant (in obvious sense) ∗-homomorphisms. Taking Kirchberg’s +KK-groups as morphisms and the same objects, we get the category KK(X). It +has a structure of a triangulated category (see [8]). For finite X, the bootstrap +class B(X) is defined as the smallest subcategory of KK(X) that is closed under +suspension, isomorphism, exact triangles, and direct sums and contains all objects +with underlying C∗-algebra C. +General methods from homological algebra suggest to study a homology the- +ory H∗ for C∗-algebras over X, taking values in some Abelian category C. Under +some mild assumptions, the machinery developed in [6, 9] yields an Adams type +spectral sequence which abuts to KK(X; , ), with an E2-term expressed in terms +of H∗. + +2000 Mathematics Subject Classification. 19K35, 46L35, 46L80, 46M18, 46M20. +The second author was supported by the German Research Foundation (Deutsche Forschungs- +gemeinschaft (DFG)) through the Institutional Strategy of the University of G¨ottingen. +1 + + +2 +RALF MEYER AND RYSZARD NEST + +For classification purposes, we need, instead of a spectral sequence, a short exact +sequence of the type (1.1): + +(1.2) +ExtC +� +H∗+1(A), H∗(B) +� +֌ KK∗(X; A, B) ։ HomC +� +H∗(A), H∗(B) +� +, + +and a precise description of the range of H∗. +In this case, given two C∗-algebras A and B over X that belong to the bootstrap +class, an isomorphism of H∗(A) to H∗(B) lifts to a KK(X)-equivalence between A +and B. The results of Eberhard Kirchberg then allow to lift this KK(X)-equivalence +to a ∗-isomorphism A ∼= B, provided A and B are tight, purely infinite, stable, +nuclear and separable; here tightness means that the maps Prim(A) → X and +Prim(B) → X are homeomorphisms (see [5]). It is also shown in [8] that, in the +case when X is finite, any object of the bootstrap class is KK(X)-equivalent to a +tight, purely infinite, stable, nuclear, separable C∗-algebra over X. +Hence the existence of an exact sequence of the form (1.2) for all objects of the +bootstrap class leads to a complete classification of the tight, purely infinite, stable, +nuclear, separable C∗-algebras over X in terms of their image under the functor H∗. + +1.2. Main results. It is relatively easy to construct filtrations on KK which pro- +duce spectral sequences which converge to KK-groups on the bootstrap category +and whose E2-term involves only the K-theory of the quotients K∗(A/J) for the +ideals J corresponding to minimal open subsets of X; an example is the filtration +used in [8, Section 4.1]. +However, this spectral sequence is not very useful for +practical purposes, since it does not degenerate at the E2-level. The second dif- +ferential involves, in particular, the K-theory of various subquotients I/J for the +ideals I ⊂ J ⊂ A and the associated six-term exact sequences in K-theory + +(1.3) + +K0 +� +I +� +� K0 +� +J +� +� K0 +� +J/I +� + +� + +K1 +� +J/I +� + +� + +K1 +� +J +� +� +K1 +� +I +� +. +� + +Also higher differentials do not vanish. +To get a short exact sequence instead, we need to consider more sophisticated ho- +mology theories. The homology theory analysed here is “filtrated K-theory,” which +is in some sense the second approximation to this spectral sequence. Roughly speak- +ing, filtrated K-theory comprises the K-theory of various subquotients together with +all canonical maps between these groups. We will make this definition precise later. +The part of it which involves the exact sequences (1.3) appeared previously in the +work of Gunnar Restorff [11] for Cuntz–Krieger algebras and of Mikael Rørdam [13] +and Alexander Bonkat [2] for extensions of C∗-algebras. The UCT theorem in the +case when the ideal structure is given by I1⊳I2⊳A was obtained by Gunnar Restorff +in his phd-thesis [12], where he introduced an invariant which is a particular case +of filtrated K-theory. +In this paper we prove the following + +Theorem 1.1. The filtrated K-theory satisfies the Universal Coefficient Theorem +and is a complete invariant for C∗-algebras over those finite topological spaces with +a totally ordered lattice of open subsets. + +Note that a C∗-algebra over a space of the type described in this result is essen- +tially the same as a C∗-algebra A together with a finite increasing chain of ideals + +{0} = I0 ⊳ I1 ⊳ I2 ⊳ I3 ⊳ · · · ⊳ In−1 ⊳ In = A. + +We will also show that the spectral sequence associated to the filtrated K-theory +does not collapse in general. Let (X, <) be the partially ordered set, where X = + + +C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY +3 + +{1, 2, 3, 4} with the partial order given by 1, 2, 3 < 4 and no further strict inequalities +between 1, 2, 3. A C∗-algebra over this space is a C∗-algebra A together with an +ideal I and a decomposition of A/I into a direct sum of three orthogonal ideals. + +Theorem 1.2. The filtrated K-theory over (X, <) does not satisfy the Universal +Coefficient Theorem and is not a complete invariant. + +In fact, we give an explicit example of two C∗-algebras A and B over X in the +bootstrap class that have isomorphic filtrated K-theory but are not KK(X)-equivalent. +However, for the particular four-point space X, we still get a complete invari- +ant and a Universal Coefficient Theorem as in (1.2), by adding another K-theory +functor to filtrated K-theory. +It is not clear how to construct such an enriched and still computable filtrated +K-theory for general finite spaces. + +1.3. The general machinery. Now we explain the general machinery behind our +approach. Let us fix a finite topological space X. The first step is the correct +definition of filtrated K-theory. The filtrated K-theory of a C∗-algebra A over X +comprises the Z/2-graded Abelian groups K∗ +� +A(Y ) +� +for all locally closed subsets +Y ⊆ X together with all natural transformations between these groups. The main +issue here is to find all natural transformations. These natural transformations +enter in the definition of the target category of the filtrated K-theory functor and +thus influence the Hom and Ext terms that we expect in the Universal Coefficient +Theorem. +We can guess some of these natural transformations. If U is a relatively open +subset of Y , then A(U) is an ideal in A(Y ), with quotient A(Y )/A(U) = A(Y \ U). +This C∗-algebra extension leads to a natural six-term exact sequence + +(1.4) + +K0 +� +A(U) +� +� K0 +� +A(Y ) +� +� K0 +� +A(Y \ U) +� + +� + +K1 +� +A(Y \ U) +� + +� + +K1 +� +A(Y ) +� +� +K1 +� +A(U) +� +. +� + +These exact sequences provide three types of natural transformations associated to +inclusions of open subsets, restriction to closed subset, and boundary maps. +An obvious source for relations between these natural transformations are morph- +isms of C∗-algebra extensions: since the six-term exact sequences in (1.4) are nat- +ural, each natural morphism of extensions provides some commuting diagrams, +which become relations between our generators. +But do these obvious generators and relations already describe all natural trans- +formations? This turns out to be the case for the spaces studied in this article—both +the positive and the negative examples. Although the authors know no counter- +examples, we do not expect this to be so in general. +The starting point for our study of filtrated K-theory is that the covariant func- +tors A �→ K∗ +� +A(Y ) +� +are representable, that is, they are of the form KK∗(X; RY , A) +for suitable C∗-algebras RY over X—these are the representing objects. Our con- +struction of RY yields commutative C∗-algebras, consisting of C0-functions on suit- +able locally closed subspaces of the order complex of the partial order on X. The +Yoneda Lemma tells us that natural transformations from K∗ +� +A(Y ) +� +to K∗ +� +A(Z) +� + +correspond to KK∗(X; RZ, RY ) ∼= K∗ +� +RY (Z) +� +. These groups are easy enough to +compute in the examples we consider, and turn out to be definable by the concrete +generators and relations mentioned above. +The natural transformations acting on filtrated K-theory form a Z/2-graded pre- +additive category NT . A (countable) module over NT is, by definition, an additive + + +4 +RALF MEYER AND RYSZARD NEST + +functor from NT to the category of (countable) Z/2-graded Abelian groups. By +construction, the filtrated K-theory of any C∗-algebra over X is such a countable +module. Let C be the category of countable NT -modules. This is an Abelian cat- +egory, and filtrated K-theory is a stable homological functor FK from the Kasparov +category KK(X) of C∗-algebras over X to C. +It is easy to check that the functor FK: KK(X) → C is universal in the notation +of [9]. General results on homological ideals in triangulated categories now pro- +duce a cohomological spectral sequence that converges towards KK∗(X; A, B) if A +belongs to the bootstrap class; its E2-term involves Extp +C +� +FK(A), FK(B) +� +. +The main issue is whether the Ext-groups Extp +C +� +FK(A), FK(B) +� +with p ≥ 2 +vanish, so that our spectral sequence degenerates to an exact sequence of the desired +form. +This amounts to checking whether FK(A) has a projective resolution of +length 1 in C. +Already for the non-Hausdorff two-point space considered in [2, 13], the cat- +egory C has infinite cohomological dimension, that is, there are objects that admit +no projective resolution of finite length. But these objects do not belong to the +range of the functor FK. If an NT -module A belongs to the range of FK, then +there are exact sequences + +(1.5) +· · · → A(U) → A(Y ) → A(Y \ U) → A(U) → · · · + +for any Y ∈ LC(X), U ∈ LC(Y ) because of (1.4). But there are NT -modules +without finite length projective resolutions. For totally ordered spaces, an object +of C has a projective resolution of length 1 if and only if it has a projective resolution +of finite length, if and only if the sequences (1.5) are exact, if and only if it is the +filtrated K-theory of some separable C∗-algebra over X, which we can take in the +bootstrap class. +For the four-point counterexample considered in Section 5, we first find a torsion- +free exact module that is not projective, and then use it to find an exact module +without projective resolutions of length 1. Then we find two non-isomorphic objects +of the bootstrap class with the same filtrated K-theory. The idea here is to consider +a certain exact triangle ΣC → A → B → C, which splits on the level of filtrated +K-theory, so that A ⊕ C and B have the same filtrated K-theory. But we can prove +in our concrete example that A ⊕ C and B are not KK(X)-equivalent. +A C∗-algebra over the four-point space X is a C∗-algebra A with a distinguished +ideal I and a direct sum decomposition of A/I as a direct sum of three orthogonal +ideals. +Since both direct sums and extensions of C∗-algebras can be classified +by filtrated K-theory, it is remarkable that the combination of both provides a +counterexample. Incidentally, the space Xop that corresponds to a C∗-algebra A +with a distinguished ideal I and a direct sum decomposition of I as a direct sum of +three orthogonal ideals also leads to a counterexample in a similar fashion. +For the four-point space X above, there is essentially just one module that ought +to be projective but is not. We can add another invariant to filtrated K-theory that +corresponds to this offending module. Since this changes our whole category, it may +lead to further offending modules, which would have to be added in a second step, +and this could, in principle, go on forever. But in the concrete case at hand, we get +projective resolutions of length 1 for all modules over the enriched filtrated K-theory. +As a result, the enriched filtrated K-theory classifies objects of the bootstrap class +over X up to KK(X)-equivalence, and it classifies purely infinite separable nuclear +stable C∗-algebras with primitive ideal space X and simple subquotients in the +bootstrap class. + +1.4. Some basic notation. We shall use the following notation from [8]: +∈∈ we write x ∈∈ C for objects of a category C as opposed to morphisms; + + +C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY +5 + +X topological space, often assumed sober (see [14]); +O(X) set of open subsets of X, partially ordered by ⊆; +LC(X) set of locally closed subsets of X; +LC(X)∗ set of connected, non-empty locally closed subsets of X; +⪯ specialisation preorder on X, defined by x ⪯ y ⇐⇒ {x} ⊆ {y} +A C∗-algebra; +Prim(A) primitive ideal space of A with hull–kernel topology; +I(A) set of closed ∗-ideals in A, partially ordered by ⊆; +C∗alg(X) category of C∗-algebras over X with X-equivariant ∗-homomorphisms +C∗sep(X) full subcategory of separable C∗-algebras over X; +KK(X) Kasparov category of C∗-algebras over X: its objects are separable +C∗-algebras over X, its set of morphisms from A to B is KK0(X; A, B); +B(X) the bootstrap class in KK(X); +iX +Y extension functor C∗alg(Y ) → C∗alg(X) or KK(Y ) → KK(X) for a +subset Y ⊆ X; +ix abbreviation for iX +{x} for x ∈ X; +rY +X restriction functor C∗alg(X) → C∗alg(Y ) or KK(X) → KK(Y ) for a +locally closed subset Y ⊆ X; +Σ suspension ΣA := C0(R, A). + +Roughly speaking, a space is sober if it can be recovered from the lattice O(X). +It is explained in [8, §2.5] why we may restrict attention to such spaces. For finite +spaces, sobriety is equivalent to the separation axiom T0, that is, two points are +equal once they have the same closure. +A C∗-algebra over X is pair (A, ψ) consisting of a C∗-algebra A and a continuous +map ψ: Prim(A) → X. If X is sober, this is equivalent to a map + +ψ∗ : O(X) → I(A), +U �→ A(U), + +that preserves finite infima and arbitrary suprema, that is, + +A +� � + +U∈F +U +� += +� + +U∈F +A(U), +A +� � + +U∈S +U +� += +� + +U∈S +A(U) = + +� + +U∈S +A(U), + +where F ⊆ O(X) is finite and S ⊆ O(X) is arbitrary. In particular, this implies +A(∅) = {0}, A(X) = A, and the monotonicity condition A(U) ⊳ A(V ) for U ⊆ V . +A ∗-homomorphism f : A → B between two C∗-algebras over X is X-equivariant +if f +� +A(U) +� +⊆ B(U) for all U ∈ O(X). +A subset Y ⊆ X is locally closed if and only if Y = U \ V for open subsets +V, U ∈ O(X) with V ⊆ U. Then we define A(Y ) := A(U)/A(V ) for a C∗-algebra A +over X; this does not depend on the choice of U and V by [8, Lemma 2.15]. +If Y ⊆ X is locally closed and A is a C∗-algebra over Y , then we extend A to a +C∗-algebra iX +Y A over X by iX +Y A(Z) := A(Y ∩Z) for Z ∈ LC(X). Conversely, we can +restrict a C∗-algebra B over X to a C∗-algebra rY +X(B) over Y by rY +XB(Z) := B(Z) +for all Z ∈ LC(Y ) ⊆ LC(X). +The category KK(X) is triangulated, with exact triangles coming either from +mapping cone triangles of X-equivariant ∗-homomorphisms or, equivalently, from +semi-split C∗-algebra extensions over X (see [7, 8]). Here an extension is called +semi-split if it splits by an X-equivariant completely positive contraction. +The bootstrap class B(X) is the localising subcategory of KK(X) generated by +the objects ixC for all x ∈ X. That is, it is the smallest class of objects containing +these generators that is closed under suspensions, KK(X)-equivalence, semi-split +extensions, and countable direct sums. + + +6 +RALF MEYER AND RYSZARD NEST + +2. Filtrated K-theory + +Let X be a finite topological space. We do not discuss filtrated K-theory for +C∗-algebras over infinite spaces here. + +Definition 2.1. For a locally closed subset Y ⊆ X, we define a functor + +FKY : KK(X) → AbZ/2, +FKY (A) := K∗ +� +A(Y ) +� +. + +Here Ab denotes the category of Abelian groups and AbZ/2 denotes the category of +Z/2-graded Abelian groups. + +For each Y ∈ LC(X), the functor FKY is stable and homological, that is, it +intertwines the suspension on KK(X) with the translation functor on AbZ/2 (this +functor shifts the grading), and if ΣC → A → B → C is an exact triangle in +KK(X)—this may, for instance, come from a semi-split extension A ֌ B ։ C— +then FKY (A) → FKY (B) → FKY (C) is an exact sequence in AbZ/2. +The functors FKY together form the filtrated K-theory functor. But the latter +also includes its target category, which we now define in a rather abstract way. + +Definition 2.2. For Y, Z ∈ LC(X), let NT ∗(Y, Z) be the Z/2-graded Abelian +group of all natural transformations FKY ⇒ FKZ. The composition of natural +transformations provides a product + +NT i(Y, Z) × NT j(W, Y ) → NT i+j(W, Z), +f, g �→ f ◦ g, + +which is associative and additive in each variable. +We let NT be the Z/2-graded category whose object set is LC and whose morph- +ism space Y → Z is NT ∗(Y, Z). The Abelian group structure on these morphism +spaces turns this into a pre-additive category. + +Definition 2.3. A module over NT is a grading preserving, additive functor +G: NT → AbZ/2. That is, it consists of a family of Z/2-graded Abelian groups +GY = (GY,0, GY,1) for Y ∈ LC(X) and product maps + +NT i(Y, Z) × GY,j → GZ,i+j +for all Y, Z ∈ LC(X), i, j ∈ Z/2; these product maps are associative, additive in +each variable, and the identity transformations in NT (Y, Y ) act identically on GY +for all Y ∈ LC(X). +Let Mod(NT ) be the category of NT -modules. The morphisms in Mod(NT ) +are the natural transformations of functors or, equivalently, families of grading +preserving group homomorphisms GY → G′ +Y that commute with the actions of NT . +Let Mod(NT )c be the full subcategory of countable modules. + +By construction, the natural transformations FKY ⇒ FKZ in NT ∗(Y, Z) induce +maps FKY (A) → FKZ(A) for all A ∈∈ KK(X). This turns +� +FKY (A) +� + +Y ∈LC(X) into +a module over NT . Furthermore, it is well-known that the K-theory of separable +C∗-algebras such as A(Y ) for A ∈∈ KK(X) is countable. + +Definition 2.4. Filtrated K-theory is the functor + +FK = (FKY )Y ∈LC(X) : KK(X) → Mod(NT )c, +A �→ +� +K∗ +� +A(Y ) +�� + +Y ∈LC(X). + +The target category Mod(NT )c is an important part of this definition because +we will compute groups of morphisms and extensions in this category. +Since A(∅) = {0} for all C∗-algebras over X, we have FK∅ = 0, so that ∅ is a +zero object of NT . Therefore, G∅ vanishes for any NT -module. +If Y ∈ LC(X) is not connected, that is, Y = Y1 ⊔ Y2 with two disjoint relat- +ively open subsets Y1, Y2 ∈ O(Y ) ⊆ LC(X), then A(Y ) ∼= A(Y1) ⊕ A(Y2) for any + + +C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY +7 + +C∗-algebra A over X. Hence FKY (A) ∼= FKY1(A) × FKY2(A). The natural trans- +formations that implement this natural isomorphism correspond to a direct sum +diagram Y ∼= Y1 ⊕ Y2 in NT . Therefore, any NT -module has GY ∼= GY1 ⊕ GY2; +here we use the fact that a functor that is additive on morphisms is also additive +on objects, even if the category in question is only pre-additive. +Since X is finite, any locally closed subset is a disjoint union of its connected +components. This corresponds to a direct sum decomposition Y ∼= +� +j∈π0(Y ) Yj +in NT . Therefore, we lose no information when we replace LC(X) by the subset +LC(X)∗ of non-empty, connected, locally closed subsets. + +2.1. The representability theorem. The representability theorem serves two +purposes. We will first use it to describe the category NT . Later, we use it to +construct geometric resolutions in KK(X). + +Theorem 2.5. Let X be a finite topological space. The covariant functors FKY for +Y ∈ LC(X) are representable, that is, there are objects RY ∈∈ KK(X) and natural +isomorphisms +KK∗(X; RY , A) ∼= FKY (A) = K∗ +� +A(Y ) +� + +for all A ∈∈ KK(X), Y ∈ LC(X). + +Before we prove this theorem in §2.2, we first describe the representing ob- +jects RY explicitly, and we use this to describe the groups of natural transformations +NT ∗(Y, Z) as K-theory groups of certain locally compact spaces. +The construction of RY requires some preparation. We equip X with the spe- +cialisation preorder ⪯ as in [8, §2.7]; recall that x ⪯ y if and only if {x} ⊆ {y}. +Since the topological space X is finite, it carries the Alexandrov topology of the +preorder ⪯, that is, a subset Y ⊆ X is open if and only if x ⪰ y ∈ Y implies x ∈ Y . +Similarly, Y ⊆ X is closed if and only if x ⪯ y ∈ Y implies x ∈ Y , and locally +closed if and only if x ⪯ y ⪯ z and x, z ∈ Y implies y ∈ Y . + +Definition 2.6. Let (X, ⪯) be a partially ordered set. Its order complex is the +geometric realisation of the simplicial set Ch(X) whose n-simplices are the chains +x0 ⪯ x1 ⪯ · · · ⪯ xn in X and whose face and degeneracy maps delete or double an +entry of the chain. + +Equivalently, Ch(X) is the classifying space of the thin category that has object +set X and a morphism x → y whenever x ⪯ y. +The order complex is the main ingredient in the construction of the representing +objects RY for Y ∈ LC(X). +The non-degenerate n-simplices in Ch(X) are the strict chains x0 ≺ · · · ≺ xn +in X. We let SX be the set of all strict chains. For each I = (x0 ≺ · · · ≺ xn) ∈ SX, +we let ∆I be a copy of ∆n; more formally, ∆I = {(t, I) | t ∈ ∆n}. We also let +∆◦ +I ⊆ ∆I be the corresponding open simplex ∆n \ ∂∆n. +The space Ch(X) is obtained from the union � +I∈SX ∆I by identifying ∆I with +the corresponding face in ∆J whenever I, J ∈ SX satisfy I ⊆ J. Thus the underly- +ing set of Ch(X) is a disjoint union + +(2.1) +Ch(X) = +� + +I∈SX +∆◦ +I. + +For I ∈ SX, let min I and max I be the (unique) minimal and maximal elements +in SX, respectively. We define two functions + +m, M : Ch(X) → X + +by mapping points in ∆◦ +I to min I and max I, respectively. This well-defines func- +tions on Ch(X) because of (2.1). + + +8 +RALF MEYER AND RYSZARD NEST + +Lemma 2.7. If Y ⊆ X is closed, then m−1(Y ) is open and M −1(Y ) is closed in +Ch(X). If Y ⊆ X is open, then m−1(Y ) is closed and M −1(Y ) is open. If Y ⊆ X +is locally closed, then m−1(Y ) and M −1(Y ) are locally closed. + +Proof. First we show that M −1(Y ) is closed if Y is closed. If I ∈ SX satisfies +max I ∈ Y , then max J ∈ Y for all J ⊆ I because max J ⪯ max I ∈ Y . Hence +∆I ⊆ M −1(Y ) once M −1(Y ) ∩ ∆◦ +I ̸= ∅, so that M −1(Y ) ∩ ∆I is closed for all +I ∈ SX; this implies that M −1(Y ) is closed. +A similar argument shows that m−1(Y ) is closed in Ch(X) if Y is open. Now +the remaining assertions follow easily because the maps m−1 and M −1 commute +with complements, unions, and intersections. +□ + +More explicitly, if Y ⊆ X is open, then m−1(Y ) is the union of the simplices SX +for all chains x0 ≺ x1 ≺ · · · ≺ xn with x0 ∈ Y and hence x0, . . . , xn ∈ Y . Thus + +m−1(Y ) = Ch(Y ) +if Y ⊆ X is open. + +Similarly, + +M −1(Y ) = Ch(Y ) +if Y ⊆ X is closed. + +Here we identify Ch(Y ) with a subcomplex of Ch(X) in the obvious way. +Let Xop be X with the topology for the reversed partial order ≻; that is, the +open subsets of Xop are the closed subsets of X, and vice versa. We may rephrase +Lemma 2.7 as follows: + +Proposition 2.8. The map (m, M): Ch(X) → Xop × X is continuous. + +Let +R := C +� +Ch(X) +� + +be the C∗-algebra of continuous functions on Ch(X). Since + +Prim R = Prim C +� +Ch(X) +� ∼= Ch(X), + +the map (m, M) turns R into a C∗-algebra over Xop × X. We abbreviate + +S(Y, Z) := m−1(Y ) ∩ M −1(Z) ⊆ Ch(X); + +this is a locally closed subset of Ch(X) by Lemma 2.7 + +Definition 2.9. We let RY be the C∗-algebra over X with + +RY (Z) := R(Y op × Z) = C0 +� +S(Y, Z) +� + +for all Y, Z ∈ LC(X); here Y op denotes Y with the subspace topology from Xop. +Equivalently, we let RY be the restriction of R to Y op × X, viewed as a C∗-algebra +over X via the coordinate projection Y op × X → X. + +We will prove the Theorem 2.5 for this choice of RY in §2.2. Taking this for +granted, we use the concrete description of RY to compute the groups of natural +transformations. By the Yoneda Lemma, natural transformations between the func- +tors FKY come from morphisms between the representing objects. More precisely, + +(2.2) +NT ∗(Y, Z) ∼= KK∗(X; RZ, RY ) ∼= FKZ(RY ) = K∗ +� +RY (Z) +� + += K∗ +� +R(Y op × Z) +� += K∗� +m−1(Y ) ∩ M −1(Z) +� += K∗� +S(Y, Z) +� +. + +By the way, the universal property of Kasparov theory says that it makes no +difference for the natural transformations FKY ⇒ FKZ whether we view these two +functors as defined on C∗sep(X) or KK(X). But since RY only represents FKY on +the level of KK(X), we get KK∗(X; RZ, RY ) and not the space of X-equivariant +∗-homomorphisms RZ → RY . + + +C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY +9 + +We describe S(Y, Z) more explicitly using the closure and boundary operations + +Z := {x ∈ X | there is z ∈ Z with x ⪯ z}, +∂Z := Z \ Z, + +�Y := {x ∈ X | there is y ∈ Y with x ⪰ y}, +�∂Y := �Y \ Y. + +Of course, Z is the closure of Z in X and �Y is the closure of Y in Xop. + +Lemma 2.10. If Y, Z ∈ LC(X), then + +S(Y, Z) = Ch(�Y ∩ Z) +� � +Ch(�Y ∩ ∂Z) ∪ Ch(�∂Y ∩ Z) +� +. + +In particular, + +S(Y, Z) = Ch(Y ∩ Z) \ Ch(Y ∩ ∂Z) +if Y is open, + +S(Y, Z) = Ch(�Y ∩ Z) \ Ch(�∂Y ∩ Z) +if Z is closed, + +S(Y, Z) = Ch(Y ∩ Z) +if Y is open and Z is closed. + +Proof. Let x0 ≺ x1 ≺ · · · ≺ xn be a strict chain in X. The interior of the corres- +ponding simplex belongs to S(Y, Z) if and only if x0 ∈ Y and xn ∈ Z. This implies +xj ∈ �Y and xj ∈ Z for all j, so that the simplex belongs to Ch(�Y ∩Z). Furthermore, +we neither have xj ∈ �∂Y ∩ Z for all j nor xj ∈ �Y ∩ ∂Z for all j because x0 ∈ Y +and xn ∈ Z. Thus the simplex belongs neither to Ch(�Y ∩ ∂Z) nor to Ch(�∂Y ∩ Z). +Conversely, if xj ∈ �Y ∩Z for all j and neither xj ∈ �∂Y ∩Z for all j nor xj ∈ �Y ∩∂Z +for all j, then some xj must belong to Y ∩ Z and some xk must belong to �Y ∩ Z. +Since Y ∩ Z is closed in �Y ∩ Z and �Y ∩ Z is open in �Y ∩ Z, this implies x0 ∈ Y and +xn ∈ Z. This shows that the interior of a simplex belongs to S(Y, Z) if and only if +it is contained in Ch(�Y ∩ Z) +� � +Ch(�Y ∩ ∂Z) ∪ Ch(�∂Y ∩ Z) +� +. +□ + +Lemma 2.10 and (2.2) yield + +NT ∗(Y, Z) ∼= K∗� +S(Y, Z) +� ∼= K∗� +Ch(�Y ∩ Z), Ch(�Y ∩ ∂Z) ∪ Ch(�∂Y ∩ Z) +� +. + +This is the K-theory of a finite CW-pair and hence is always finitely generated as +an Abelian group. +If C is any finite simplicial complex, then its barycentric subdivision is of the +form Ch(X), where X is the partially ordered set of non-degenerate simplices in C. +Thus NT ∗(X, X) = K∗(|C|), so that any finitely generated Abelian group arises +as NT ∗(X, X). As a consequence, special properties of the pre-additive category +NT can only be hidden in its composition. +When we identify NT ∗(Y, Z) ∼= KK∗(X; RZ, RY ), then the composition of nat- +ural transformations corresponds to the Kasparov composition product. This gets +somewhat obscured when we follow the isomorphisms + +KK∗(X; RZ, RY ) ∼= K∗ +� +RY (Z) +� += K∗� +S(Y, Z) +� +. + +To describe the composition of natural transformations in terms of K∗� +S(Y, Z) +� +, we +must first lift elements of K∗� +S(Y, Z) +� +back to KK∗(X; RZ, RY ) and then compose +them. The lifting requires a formula for the natural isomorphism + +(2.3) +KK∗(X; RY , A) → K∗ +� +A(Y ) +� + +that occurs in the Representability Theorem. By the Yoneda Lemma, any such +natural transformation is of the form f �→ f∗(ξY ) for a unique + +ξY ∈ K0 +� +RY (Y ) +� += K0� +S(Y, Y ) +� += K0� +Ch(Y ) +� +. + +The natural transformation in (2.3) is generated by the class of the 1-dimensional +trivial vector bundle over the compact space Ch(Y ) or, equivalently, the class of +the unit element in K0 +� +RY (Y ) +� +. + + +10 +RALF MEYER AND RYSZARD NEST + +In the examples we consider later, all natural transformations turn out to be +products of obvious ones, coming from the K-theory six-term exact sequences (1.4). +To check this, we only have to verify that a given element α of KK∗(X; RZ, RY ) lifts +a given element of K∗� +S(Y, Z) +� +. The isomorphism (2.3) maps α to [ξZ]⊗RZ(Z)α(Z) +in K∗ +� +RY (Z) +� += K∗� +S(Y, Z) +� +, where α(Z) in KK∗ +� +RZ(Z), RY (Z) +� +is obtained +from α by restriction to Z. This product is easy to compute. +To get acquainted with this approach to natural transformations, we compute +some important examples. Let Y ∈ LC(X) and U ∈ O(Y ). Since R is a C∗-algebra +over Xop × X, there is an extension + +(2.4) +RY \U ֌ RY ։ RU + +of C∗-algebras over X. It contains C∗-algebra extensions + +RY \U(Z) ֌ RY (Z) ։ RU(Z) + +for all Z ∈ LC(X). Let Z := Y \ U. The extension (2.4) is semi-split in C∗alg(X) +and hence has a class in KK1(X; RU, RZ) and produces an exact triangle + +(2.5) +ΣRU → RZ → RY → RU + +in KK(X). + +Lemma 2.11. The maps in the extension triangle (2.5) correspond to the natural +transformations FKU[1] ⇐ FKZ ⇐ FKY ⇐ FKU in (1.4). + +Proof. The natural transformation µY +U : FKU ⇒ FKY in (1.4) is induced by the +natural ∗-homomorphism j : A(U) → A(Y ). For A = RU, this map is invertible +because S(U, Y ) = S(U, U) = Ch(U). Hence j(ξU) ∈ K0� +S(U, Y ) +� +is again the +class of the trivial vector bundle on Ch(U); this class corresponds to the natural +transformation µY +U . The restriction map RY ։ RU in (2.4) maps [ξY ] to [ξU]— +recall that both [ξY ] and [ξU] are trivial vector bundles. Hence the restriction map +RY ։ RU and the natural transformation µZ +Y correspond to the same class—the +1-dimensional trivial vector bundle on Ch(U)—in K0� +S(U, Y ) +� +. +Similarly, the natural transformation µZ +Y : FKY ⇒ FKZ is induced by the nat- +ural ∗-homomorphism p: A(Y ) ։ A(Z). +For A = RY , this is the restriction +∗-homomorphism C +� +Ch(Y ) +� +→ C +� +Ch(Z) +� +because S(Y, Y ) = Ch(Y ) and S(Y, Z) = +Ch(Z). Since the restriction of a trivial bundle remains trivial, µZ +Y corresponds +to the trivial 1-dimensional vector bundle on S(Y, Z) = Ch(Z). The embedding +RZ ։ RY restricts to an identity map on Z because S(Z, Z) = S(Z, Y ) = Ch(Z). +Since this maps [ξZ] to the trivial bundle, the embedding RZ ։ RY and µZ +Y both +correspond to the same class—the 1-dimensional trivial vector bundle on Ch(Z)—in +K0� +S(Y, Z) +� +. +Finally, we study the boundary map δU +Z : FKZ ⇒ FKU[1]. +We claim that it +corresponds to the class of the extension RZ ֌ RY ։ RU in KK1(X; RU, RZ). +To prove this, we use that Ch(Y ) is the join of the spaces Ch(U) and Ch(Z), so +that there is a continuous map f : Ch(Y ) → [0, 1] whose fibres over 0 and 1 are +Ch(U) and Ch(Z), respectively. +More precisely, let x0 ≺ x1 ≺ · · · ≺ xn be a strict chain in Y and let ξ be a point +of the corresponding simplex with coordinates (t0, . . . , tn) with t0 + · · · + tn = 1, +that is, ξ = t0x0 + · · · + tnxn. Then there is j ∈ {0, . . . , n} with x0, . . . , xj ∈ U, +xj+1, . . . , xn ∈ Z. We can, therefore, write ξ = tUξU + tZξZ with + +ξU = t0x0 + · · · + tjxj + +tU +∈ Ch(U), +tU = t0 + · · · + tj, + +ξZ = tj+1xj+1 + · · · + tnxn + +tZ +∈ Ch(Z), +tZ = tj+1 + · · · + tn. + + +C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY +11 + +We define a continuous map f : Ch(Y ) → [0, 1] by ξ �→ tZ. We have + +S(U, U) = Ch(U) = f −1(0), +S(Z, Z) = Ch(Z) = f −1(1) + +by construction, and hence + +S(Z, U) = Ch(Y ) \ +� +Ch(U) ⊔ Ch(Z) +� += f −1� +(0, 1) +� +. + +Now we can compute some boundary maps. The boundary map + +K0� +S(Z, Z) +� ∼= K0 +� +RZ(Z) +� +→ K1 +� +RZ(U) +� ∼= K1� +S(Z, U) +� + +maps the class of the trivial bundle [ξZ] to f ∗(δ), where δ denotes a generator +of Z ∼= K1� +(0, 1) +� +; this follows from the naturality of the boundary map. +The +boundary map + +K0� +S(U, U) +� ∼= K0 +� +RU(U) +� +→ K1 +� +RZ(U) +� ∼= K1� +S(Z, U) +� + +for the extension RZ ֌ RY ։ RU maps the class of the trivial bundle [ξU] to +−f ∗(δ), again by naturality of the boundary map. +□ + +Remark 2.12. The proof also describes the classes in K0� +S(U, Y ) +� +, K0� +S(Y, Z) +� +, +and K1� +S(Z, U) +� +that correspond to the natural transformations in (1.4). +The +natural transformations FKU ⇒ FKY and FKY ⇒ FKZ are represented by the +classes of the trivial vector bundles over the compact spaces S(U, Y ) and S(Y, Z); +the natural boundary map FKZ ⇒ FKU[1] is represented by f ∗(δ) for a generator +of K1� +(0, 1) +� +. + +2.2. Proof of Theorem 2.5. We check first that the natural transformation +KK∗(X; RY , A) → K∗ +� +A(Y ) +� +induced by ξY is an isomorphism if Y is the min- +imal open subset Ux containing some point x ∈ X. The adjointness relation + +KK∗(X; ix(A), B) ∼= KK∗ +� +A, B(Ux) +� + +for all B ∈∈ KK(X) established in [8, Proposition 3.12] yields + +KK∗(X; ix(C), B) ∼= KK∗ +� +C, B(Ux) +� += FKUx(B), + +that is, ix(C) represents FKUx. To check that RUx does so as well, we must show +that ix(C) and RUx are KK(X)-equivalent. + +Recall that ix(C) = (C, x), where x denotes the map Prim(C) ∼= {x} +⊆ +−→ X, and + +ix(C)(Z) = + +� +C +if x ∈ Z, +0 +otherwise + +for all Z ∈ LC(X). +Since Ux = {y ∈ X | x ⪯ y}, the preordered set Ux has a minimal point, +namely x. +Therefore, the space Ch(Ux) is starlike and hence contractible in a +canonical way towards x. The path from a point in ∆I for I ∈ SUx to the base +point in ∆x lies in ∆I∪{x}. Since max I ∪ {x} = max I, the contraction preserves +the ideals RUx(V ) for V ∈ O(X), so that we get a homotopy equivalence between +C +� +Ch(Ux) +� +and ix(C) in C∗alg(X). Thus RUx corepresents FKUx as well. It is easy +to see that the natural isomorphism KK∗(X; RUx, ) ∼= FKUx is induced by ξUx. +Let Good ⊆ LC(X) be the set of all Z ∈ LC(X) for which the natural trans- +formation KK∗(X; RZ, A) → FKZ(A) induced by ξZ is an isomorphism. We must +show Good = LC(X). We have just seen that Ux ∈ Good for all x ∈ X. +Let Y ∈ LC(X) and U ∈ O(Y ); we claim that all three of U, Y , and Y \ U +are good once two of them are. This follows from the Five Lemma because the + + +12 +RALF MEYER AND RYSZARD NEST + +maps induced by ξZ for Z = U, Y, Y \ U intertwine the maps in the six-term exact +sequences (1.4) and + +KK0(X; RU, A) +� KK0(X; RY , A) +� KK0(X; RY \U, A) + +� + +KK1(X; RY \U, A) + +� + +KK1(X; RY , A) +� +KK1(X; RU, A) +� + +for any A ∈∈ KK(X); the latter six-term exact sequence is induced by the semi- +split extension (2.5). The commutativity of the relevant diagrams follows from the +computations in the proof of Lemma 2.11 (which do not depend on Theorem 2.5). +The two-out-of-three property of Good implies: + +U, V ∈ O(X), +U, V, U ∩ V ∈ Good +=⇒ +U ∪ V ∈ Good + +because (U ∪ V ) \ U = V \ (U ∩ V ). By induction on the length of U, this implies +that all open subsets of X belong to Good. Since any locally closed subset is a +difference of two open subsets, we conclude that Good = LC(X). This finishes the +proof of Theorem 2.5. + +3. An example + +In this section, we restrict our attention to a special class of spaces, namely, the +spaces X = {1, . . ., n} totally ordered by ≤ for n ∈ N. We let + +[a, b] := {x ∈ X | a ≤ x ≤ b}. + +for a, b ∈ Z. We equip X with the Alexandrov topology, so that the open subsets +are [a, n] for all a ∈ X; the closed subsets are [1, b] with b ∈ X, and the locally +closed subsets are those of the form [a, b] with a, b ∈ X and a ≤ b. Any locally +closed subset of X is connected. + +3.1. Computations with the order complex. Since any subset of X is totally +ordered, the space Ch([a, b]) is just a closed simplex of dimension b − a for any +b ≥ a. We denote the corresponding face of Ch(X) by ∆[a,b]. This is understood +to be empty for a > b. +From now on, we let + +Y = [a1, b1], +Z = [a2, b2], +with 1 ≤ a1 ≤ b1 ≤ n and 1 ≤ a2 ≤ b2 ≤ n. + +Then �Y = [a1, n], �∂Y = [b1 + 1, n], Z = [1, b2], and ∂Z = [1, a2 − 1]. Lemma 2.10 +yields + +S(Y, Z) = ∆[a1,b2] \ +� +∆[a1,a2−1] ∪ ∆[b1+1,b2] +� +. + +Now we distinguish three cases: + +Case 1: If a2 ≤ a1 ≤ b2 ≤ b1, then S(Y, Z) = ∆[a1,b2] is a non-empty closed simplex. +Hence NT ∗(Y, Z) ∼= K∗� +S(Y, Z) +� ∼= Z[0] (this means Z in degree 0). +Case 2: If a2 − 1 ≤ b1, a1 < a2, and b1 < b2, then S(Y, Z) is obtained from a closed +simplex by removing two disjoint, non-empty closed faces. Excision yields +NT ∗(Y, Z) ∼= K∗� +S(Y, Z) +� ∼= Z[1] (this means Z in degree 1). +Case 3: In all other cases, S(Y, Z) is either empty, a difference of two closed sim- +plices, or a difference σ \ (τ1 ∪ τ2) for two non-empty closed faces τ1 and τ2 +of a simplex σ that intersect. Then τ1 ∪ τ2 and σ are both contractible, so +that NT ∗(Y, Z) ∼= K∗� +S(Y, Z) +� ∼= 0. + + +C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY +13 + +Summing up, we get + +(3.1) +NT ∗(Y, Z) = + + + + + + + + +Z[0] +if a2 ≤ a1 ≤ b2 ≤ b1, +Z[1] +if a2 − 1 ≤ b1, a1 < a2, and b1 < b2, +0 +otherwise. + +3.2. Products of natural transformations. Our next task is to identify the +natural transformations that correspond to the generators of the groups in (3.1); +this also allows us to compute products in NT . +First we study the grading preserving transformations that appear in the first +case. We introduce a partial order ≥ and a strict partial order ≫ on LC(X) by + +[a1, b1] ≥ [a2, b2] +⇐⇒ +a1 ≥ a2 and b1 ≥ b2, + +[a1, b1] ≫ [a2, b2] +⇐⇒ +a1 > b2. + +Our computation shows that NT 0(Y, Z) ̸= {0} if and only if Y ≥ Z but not +Y ≫ Z. This is equivalent to Y ∩ Z being non-empty, closed in Y , and open in Z. +Under these assumptions, there is a natural non-zero ∗-homomorphism given by +the composition +µZ +Y : A(Y ) ։ A(Y ∩ Z) ֌ A(Z) +because A(Y ∩ Z) is a quotient of A(Y ) and an ideal in A(Z). The natural trans- +formation FKY ⇒ FKZ induced by µZ +Y maps ξY ∈ FKY,0(RY ), which is the class +of the trivial line bundle over S(Y, Y ) = ∆[a1,b1], to the trivial line bundle over +S(Y, Z) = ∆[a1,b2]. Since this is the generator of FKZ,0(RY ) = K0� +S(Y, Z) +� ∼= Z[0], +the natural transformation µZ +Y generates NT 0(Y, Z). +If Y ≫ Z, then we let µZ +Y : A(Y ) → A(Z) be the zero map, which induces the zero +transformation FKY ⇒ FKZ. With this convention, we get µZ +Y ◦ µY +W = µZ +W for all +Y, Z, W ∈ LC(X) with W ≥ Y ≥ Z, also if W ≫ Z; this equation holds on the level +of ∗-homomorphisms and, therefore, also for the induced natural transformations. +We can sum this up as follows: + +Lemma 3.1. The category NT 0 of grading-preserving natural transformations +FKY ⇒ FKZ for Y, Z ∈ LC(X) is the pre-additive category generated by natural +transformations µZ +Y : FKY ⇒ FKZ for all Y ≥ Z with the relations µZ +Y ◦ µY +W = µZ +W +for W ≥ Y ≥ Z and µZ +Y = 0 for Y ≫ Z. + +This list of generators is longer than necessary. Clearly, we can write any µZ +Y +as a product of the transformations µ[a−1,b] +[a,b] +for 2 ≤ a ≤ b ≤ n and µ[a,b−1] +[a,b] +for +1 ≤ a < b ≤ n. Moreover, these transformations themselves are indecomposable, +that is, they cannot be written themselves as products in a non-trivial way. +Now we turn to the natural transformations of degree 1. For any b ∈ X and any +C∗-algebra A over X, we have a natural C∗-algebra extension + +A([b, n]) ֌ A([1, n]) ։ A([1, b − 1]), + +which generates an odd natural transformation + +δb : FK[1,b−1] ⇒ FK[b,n]. + +Composing with the grading preserving natural transformations µ above, we get a +natural transformation of degree 1 + +(3.2) +δZ +Y : FKY = FK[a1,b1] +µ=⇒ FK[1,a2−1] +δa2 +==⇒ FK[a2,n] +µ=⇒ FK[a2,b2] = FKZ + +whenever b1 ≥ a2 − 1. +Equation (3.1) predicts that this transformation vanishes if a1 ≥ a2 or b1 ≥ b2. +This can be verified as follows. Vanishing for a1 ≥ a2 is clear because then [a1, b1] ≫ +[1, a2−1]. By the naturality of the boundary map, the transformation in (3.2) agrees + + +14 +RALF MEYER AND RYSZARD NEST + +with the composition of µ: FK[a1,b1] ⇒ FK[a1,a2−1] with the boundary map for the +extension + +(3.3) +A([a2, b2]) ֌ A([a1, b2]) ։ A([a1, a2 − 1]). + +If b1 ≥ b2, then µ[a1,a2−1] +[a1,b1] +factors through the quotient map in (3.3). +But the +composite of two maps in a six-term exact sequence vanishes. +Equation (3.2) produces a natural transformation δZ +Y ∈ NT 1(Y, Z) whenever +a1 < a2, b1 < b2, and a2 − 1 ≤ b1, that is, whenever (3.1) predicts NT 1(Y, Z) +to be non-zero. We claim that δZ +Y generates this group. This follows because the +natural transformation δZ +Y maps the class of the trivial line bundle over S(Y, Y ) to +the generator of K1� +S(Y, Z) +� ∼= Z. +Notice that NT 1([a2, n], Z) = {0} for any Z ∈ LC(X). Since the natural trans- +formation (3.2) above factors through FK[a2,n], any product of two odd natural +transformations vanishes. Thus the category NT is a split extension of NT 0 by +the bimodule NT 1. The bimodule structure on NT 1 is very simple: a product +µZ +Y ◦ δY +W or δZ +Y ◦ µY +W is equal to δZ +W whenever all three natural transformations are +defined, and zero otherwise. + +Example 3.2. To make our constructions more concrete, we now consider the ex- +ample n = 2, which corresponds to extensions of C∗-algebras. There are only three +non-empty locally closed subsets: 1 = [1, 1], 12 = [1, 2], and 2 = [2, 2]. The order +complex is an interval; we label its end points 1 and 2. The map (m, M) from +Ch(X) = [1, 2] to Xop × X maps + +1 �→ (1, 1), +2 �→ (2, 2), +]1, 2[ �→ (1, 2). + +Correspondingly, we have + +S(1, 1) = {1}, +S(1, 2) = ]1, 2[, +S(1, 12) = [1, 2[, + +S(2, 1) = ∅, +S(2, 2) = {2}, +S(2, 12) = {2}, + +S(12, 1) = {1}, +S(12, 2) = ]1, 2], +S(12, 12) = [1, 2]. + +Taking K-theory, we get + +NT (1, 1) = Z[0], +NT (1, 2) = Z[1], +NT (1, 12) = 0, + +NT (2, 1) = 0, +NT (2, 2) = Z[0], +NT (2, 12) = Z[0], + +NT (12, 1) = Z[0], +NT (12, 2) = 0, +NT (12, 12) = Z[0]. + +3.3. Ring-theoretic properties of the natural transformations. We now ob- +serve some general ring-theoretic properties of NT for X = {1, . . . , n} with the +total order. We exclude the trivial case n = 1. We may replace NT by a Z/2- +graded ring by taking the direct sum of NT ∗(Y, Z) for all Y, Z ∈ LC(X)∗ and +defining the product as usual for a category ring. Then NT -modules become Z/2- +graded modules over this Z/2-graded ring, and ring-theoretic notions such as the +Jacobson radical and the balanced tensor product ⊗NT make sense. + +Definition 3.3. Let NT nil ⊆ NT be the subgroup spanned by the natural trans- +formations µZ +Y with Y ̸= Z and δZ +Y with arbitrary Y, Z. +Let NT ss ⊆ NT be the subgroup spanned by the natural transformations µY +Y +with Y ∈ LC(X)∗. + +Lemma 3.4. The subgroup NT nil is the maximal nilpotent ideal in NT , it is the +nilradical and the Jacobson radical of NT . The subgroup NT ss is a semi-simple +subring, and NT decomposes as a semi-direct product NT nil ⋊ NT ss. + +Proof. Since all µY +Y are idempotent, NT ss is a subring isomorphic to ZLC(X)∗ with +pointwise multiplication. It is easy to see that NT nil is an ideal in NT . It is + + +C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY +15 + +nilpotent, that is, NT k +nil = {0} for some k ∈ N, because LC(X)∗ is finite and ≥ +is a partial order on it. Since NT = NT nil ⊕ NT ss as Abelian groups, we get +the desired semi-direct product decomposition. Since the Jacobson radical of NT ss +vanishes, NT nil is both the nilradical and the Jacobson radical of NT . +□ + +We are going to use Lemma 3.4 to characterise the projective NT -modules. This +characterisation involves the following two definitions. + +Definition 3.5. We call an NT -module M exact if the chain complexes + +· · · → M(U) +µY +U +−−→ M(Y ) +µY \U +Y +−−−→ M(Y \ U) +δU +Y \U +−−−→ M(U) → · · · + +are exact for all Y ∈ LC(X), U ∈ O(Y ) as in (1.5). + +Proposition 3.6. Let K ֌ E ։ Q be an extension of NT -modules. If two of the +modules K, E, Q are exact, so is the third one. + +Proof. Given U and Y as above and a module M, let C•(M) be the chain complex + +· · · → M(U)[m] → M(Y )[m] → M(Y \ U)[m] → M(U)[m − 1] → · · · . + +Then C•(K) ֌ C•(E) ։ C•(Q) is an extension of chain complexes. The long +exact homology sequence shows that all three of these chain complexes are exact +once two of them are exact. +□ + +Definition 3.7. Given an NT -module M, we let + +NT nil · M = {x · m | x ∈ NT nil, m ∈ M}, +Mss := M/NT nil · M. + +We call Mss the semi-simple part of M. + +Since the tensor product over NT is right exact, Mss ∼= NT ss ⊗NT M. We need +the following more concrete description of Mss or, equivalently, of NT nil · M. + +Lemma 3.8. Let M be an NT -module and let Y = [a, b] with 1 ≤ a ≤ b ≤ n. +Then + +(NT nil · M)(Y ) = + + + + + + + + + + + + + + + + + + +µY +[a+1,b](M[a + 1, b]) + µY +[a,b+1](M[a, b + 1]) +if a < b < n, + +µY +[a,b+1](M[a, b + 1]) +if a = b < n, + +µY +[a+1,b](M[a + 1, b]) + δY +[1,a−1](M[1, a − 1]) +if 1 < a < b = n, + +µY +[a+1,b](M[a + 1, b]) +if 1 = a < b = n, + +δY +[1,a−1](M[1, a − 1]) +if a = b = n. + +If M is exact, then + +(NT nil · M)(Y ) = + +� +ker +� +δ[a+1,b+1] +[a,b] +: M[a, b] → M[a + 1, b + 1] +� +if b < n, + +ker +� +µ[1,a] +[a,b] : M[a, b] → M[1, a] +� +if b = n. + +Proof. The first assertion holds because any natural transformation FKZ ⇒ FKY +with Z ̸= Y factors through µY +[a+1,b] or µY +[a,b+1] if a < b < n, through µY +[a,b+1] if + +a = b < n, and so on. Here we use that the natural transformations µ[a−1,b] +[a,b] +for + +2 ≤ a ≤ b ≤ n, µ[a,b−1] +[a,b] +for 1 ≤ a < b ≤ n, and δ[a,n] +[1,a−1] for 2 ≤ a ≤ n already +generate NT ∗, that is, all other transformations µZ +Y or δZ +Y with Y ̸= Z can be +written as products of these generators. By the way, these natural transformations +even form a basis for the subquotient NT nil/NT 2 +nil. +Now assume that M is exact. If a = b < n, then + +(NT nil · M)[a, a] = range +� +µ[a,a] +[a,a+1] +� += ker +� +δ[a+1,a+1] +[a,a] +� +. + + +16 +RALF MEYER AND RYSZARD NEST + +Similarly, we get + +(NT nil · M)[n, n] = ker +� +µ[1,n] +[n,n] +� +, +(NT nil · M)[1, n] = ker +� +µ[1,1] +[1,n] +� +. + +Given f1 : A1 → B and f2 : A2 → B and two exact sequences + +A1 +f1 +−→ B +g1 +−→ C1, +A2 +g1f2 +−−−→ C1 +g2 +−→ C2, + +we have + +(3.4) +range(f1) + range(f2) = ker(g1) + range(f2) + += {x ∈ B | g1(x) ∈ range(g1 ◦ f2) = ker(g2)} = ker(g2 ◦ g1). + +If a < b < n, then we apply this to the maps on M induced by f1 = µY +[a+1,b] +and f2 = µY +[a,b+1] with Y = [a, b]. +We get g1 = µ[a,a] +Y +, g1 ◦ f2 = µ[a,a] +[a,b+1], and + +hence g2 = δ[a+1,b+1] +[a,a] +and g2 ◦ g1 = δ[a+1,b+1] +[a,b] +. This yields the desired formula for +(NT nil · M)[a, b] for a < b < n, using the exactness of M. If a < b = n, then we +apply the same reasoning to f1 = µY +[a+1,b] and f2 = δY +[1,a−1]. Here we get g1 = µ[a,a] +Y +as above, g1 ◦ f2 = δ[a,a] +[1,a−1], and hence g2 = µ[1,a] +[a,a] and g2 ◦ g1 = µ[1,a] +[a,b]. This yields +the desired formula for (NT nil · M)[a, b] for a < b = n. +□ + +Remark 3.9. The natural transformation δ[a+1,b+1] +[a,b] +for b < n or µ[1,a] +[a,n] for b = n +is the longest natural transformation out of [a, b] in the following sense: it factors +through δZ +[a,b] or µZ +[a,b] whenever the latter is defined and non-zero. Thus Lemma 3.8 +identifies NT nil·M(Y ) with the largest proper subgroup of M(Y ) that is the kernel +of some δZ +[a,b] or µZ +[a,b]. + +The following proposition is a rather trivial variant of the Nakayama Lemma. +Unlike in the usual Nakayama Lemma, we do not assume the module to be finitely +generated. This is no problem because the relevant ideal NT nil is nilpotent. + +Proposition 3.10. Let M be an NT -module with Mss = 0. Then M = 0. + +Proof. By assumption, M = NT nil · M. By induction, this implies M = NT j +nil · M +for all j ∈ N. Since NT k +nil = 0 for some k, we get M = 0. +□ + +3.4. Characterisation of free and projective modules. + +Definition 3.11. For Y ∈ LC(X), the free NT -module on Y is defined by + +PY (Z) := NT ∗(Y, Z) +for all Z ∈ LC(X). + +An NT -module is called free if it is isomorphic to a direct sum of degree-shifted +free modules PY [j], j ∈ Z/2. + +Theorem 3.12. Let M be an NT -module. Then the following are equivalent: + +(i) M is a free NT -module. +(ii) M is a projective NT -module. +(iii) Mss(Y ) = NT ss ⊗NT M(Y ) is a free Abelian group for all Y ∈ LC(X) and +TorNT +1 +(NT ss, M) = 0. +(iv) M(Y ) is a free Abelian group for all Y ∈ LC(X) and M is exact. + +Here TorNT +1 +denotes the first derived functor of ⊗NT . The first three conditions +remain equivalent when we replace NT by any ring that is a nilpotent extension of +the ring ZN for some N ∈ N. + + +C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY +17 + +Proof. The Yoneda Lemma asserts that Hom(PY , M) ∼= M(Y ) for all Y ∈ LC(X) +and all NT -modules M. Hence free modules are projective, that is, (1)=⇒(2). A +functor of the form M �→ R ⊗S M for a ring homomorphism S → R always maps +free modules to free modules and hence maps projective modules to projective mod- +ules. Furthermore, derived functors like TorNT +1 +automatically vanish on projective +modules. This yields the implication (2)=⇒(3). We are going to prove that (3) +implies (1). +Since Mss(Y ) is a free Abelian group for all Y , Mss is a free module over NT ss ∼= +ZLC(X)∗. Hence P := NT ⊗NT ssMss is a free NT -module. The canonical projection +M → Mss splits by an NT ss-module homomorphism because Mss is free. This +induces an NT -module homomorphism f : P → M because of the adjointness +relation +HomNT (NT ⊗NT ss X, Y ) ∼= HomNT ss(X, Y ). + +We claim that f is invertible, so that M ∼= P is a free module as asserted. We have + +Pss = NT ss ⊗NT NT ⊗NT ss Mss ∼= NT ss ⊗NT ss Mss ∼= Mss. + +Inspection shows that this isomorphism is induced by f. Since the functor M �→ Mss +is right-exact, this implies coker(f)ss = 0 and hence coker(f) = 0 by the Nakayama +Lemma (Proposition 3.10). That is, f is an epimorphism. +Let K := ker(f), then we get an exact sequence of NT -modules K ֌ P ։ M. +The derived functors of NT ss ⊗NT +provide a long exact sequence + +(3.5) +0 → TorNT +1 +(NT ss, M) → Kss → Pss +f−→ +∼ += Mss → 0. + +This exact sequence ends at TorNT +1 +(NT ss, P) = 0 because P is projective. Since +TorNT +1 +(NT ss, M) = 0 by assumption, we conclude that Kss = 0. Hence another +application of the Nakayama Lemma shows that ker(f) = 0 as well. Thus f is +invertible. This finishes the proof of the implication (3)=⇒(1), showing that the +first three conditions are equivalent. Furthermore, our argument so far works for any +split nilpotent extension of ZN for some N ∈ N because this is the only information +about NT that we have used. Nilpotent extensions of the ring ZN always split +because we can lift orthogonal idempotents in nilpotent extensions. +Free NT -modules are exact, and they consist of free Abelian groups by (3.1). +This yields the implication (1)=⇒(4). We are going to prove that (4) implies (3). +This will finish the proof of the theorem. Since we will use this once again later, +we state half of this argument as a separate lemma: + +Lemma 3.13. Let M be an exact NT -module. Then TorNT +1 +(NT ss, M) = 0. + +Proof. Let π: P → M be an epimorphism with a projective NT -module P, and +let K := ker π. Since projective modules are exact and K ֌ P ։ M is a module +extension, Proposition 3.6 shows that K is exact. We still have an exact sequence +as in (3.5). +Since K and P are exact, Lemma 3.8 identifies Kss(Y ) and Pss(Y ) in a natural +way with subspaces of K(Z) and P(Z) for suitable Z; here we use A/ ker(f) ∼= +range(f) for a group homomorphism f : A → B. Since the map K(Z) → P(Z) +is injective, so is the map Kss(Y ) → Pss(Y ). +Hence the map Kss → Pss is a +monomorphism, forcing TorNT +1 +(NT ss, M) = 0 by (3.5). +□ + +To finish the proof of the implication (4)=⇒(3) in Theorem 3.12, it remains to +check that Mss(Y ) is free for all Y if M is exact and M(Y ) is free for all Y . We +use Lemma 3.8 once again to describe Mss(Y ) as the range of a canonical element +in NT ∗(Y, Z) for a suitable Z. Thus Mss(Y ) is isomorphic to a subgroup of M(Z), +which is a free group by assumption. Hence Mss(Y ) is free as well. +□ + + +18 +RALF MEYER AND RYSZARD NEST + +4. Homological algebra in KK(X) + +Let X be a sober topological space. We are going to apply to KK(X) the gen- +eral machinery for doing homological algebra in triangulated categories discussed +in [9]. This theory goes back to the work on relative homological algebra by Samuel +Eilenberg and John Coleman Moore ([4]), which was carried over to the setting of +triangulated categories by Daniel Christensen [3] and Apostolos Beligiannis [1]. + +4.1. An ideal in KK(X). Our starting point is a rough idea of the invariant we +want to use. This rough idea is expressed by a homological ideal in the triangulated +category. The ideal I in KK(X) relevant for us is defined by + +(4.1) +I(A, B) := +� +f ∈ KK(X; A, B) +�� + +f∗ : K∗ +� +A(Y ) +� +→ K∗ +� +B(Y ) +� +vanishes for all Y ∈ LC(X) +� +. + +It makes no difference if we use LC(X) or LC(X)∗ here. +We claim that I is a homological ideal in the triangulated category KK(X); that +is, it is the kernel (on morphisms) of a stable homological functor from KK(X) +to some stable Abelian category; stability means that the functor intertwines the +suspension automorphism on KK(X) with a given suspension automorphism on the +target Abelian category. +Our starting point is a bare form of filtrated K-theory. Recall the functors + +FKY : KK(X) → AbZ/2, +A �→ K∗ +� +A(Y ) +� + +for Y ∈ LC(X) from Definition 2.1 and let + +F := (FKY )Y ∈LC(X) : KK(X) → +� + +Y ∈LC(X)∗ +AbZ/2, +A �→ +� +K∗ +� +A(Y ) +�� + +Y ∈LC(X)∗. + +The target category � +Y ∈LC(X)∗ AbZ/2 of F is Abelian and carries an obvious sus- +pension functor that shifts the Z/2-grading. The functor F is a stable homological +functor, that is, it intertwines the suspension automorphisms and maps exact tri- +angles to long exact sequences. By definition, + +(4.2) +I = +� + +Y ∈LC(X)∗ +ker FKY = ker F, + +that is, f ∈ I(A, B) if and only if F(f) = 0. Hence I is a homological ideal with +defining functor F. +We also have I = ker FK with FK as in Definition 2.4: the two functors F +and FK only differ through their target categories. For the time being, we pretend +that we do not yet know anything about filtrated K-theory beyond the ideal I it +defines. The general machinery will automatically lead us to the functor FK. +As explained in [9], the homological ideal I yields various notions of homological +algebra. The following descriptions of these notions follow from [9, Lemmas 3.2 +and 3.9, Definition 3.21]. +• A morphism f ∈ KK∗(X; A, B) is +– I-epic if the induced maps K∗ +� +A(Y ) +� +→ K∗ +� +B(Y ) +� +are surjective for +all Y ∈ LC(X); +– I-monic if the induced maps K∗ +� +A(Y ) +� +→ K∗ +� +B(Y ) +� +are injective for +all Y ∈ LC(X); +– an I-equivalence if the induced maps K∗ +� +A(Y ) +� +→ K∗ +� +B(Y ) +� +are +bijective for all Y ∈ LC(X). +• A homological functor F : KK(X) → C to some Abelian category C is +I-exact if F(f) = 0 for all f ∈ I; equivalently, F maps I-epimorphisms +to epimorphisms or F maps I-monomorphisms to monomorphisms. + + +C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY +19 + +• An object A ∈∈ KK(X) is +– I-contractible if K∗ +� +A(Y ) +� += 0 for all Y ∈ LC(X); +– I-projective if the functor KK∗(X; A, ) is I-exact; equivalently, I(A, B) = +0 for all B ∈∈ KK(X), or: any I-epimorphism B → A splits (see [9] +for more equivalent characterisations). +• A chain complex + +· · · → An+1 +δn+1 +−−−→ An +δn +−→ An−1 +δn−1 +−−−→ An−2 → · · · + +in KK(X)—that is, An ∈∈ KK(X) and δn ∈ KK(X; An, An−1) for all n ∈ Z, +subject to the condition δn−1 ◦ δn = 0—is I-exact (in some degree n) if the +induced chain complexes of Z/2-graded Abelian groups + +· · · → K∗ +� +An+1(Y ) +� (δn+1)∗ +−−−−−→ K∗ +� +An(Y ) +� (δn)∗ +−−−→ K∗ +� +An−1(Y ) +� +→ · · · + +are exact (in degree n) for all Y ∈ LC(X). +• An I-projective resolution of A ∈∈ KK(X) is an I-exact chain complex + +· · · → P2 +δ2 +−→ P1 +δ1 +−→ P0 +δ0 +−→ A → 0 → · · · + +with I-projective entries Pn for all n ∈ N. +We shall soon see that there are enough I-projective objects in the sense that any +object of KK(X) has an I-projective resolution. Such resolutions are unique up to +chain homotopy equivalence once they exist. +We use projective resolutions to define derived functors (see [9, Definition 3.27]): +just apply the functor to be derived to an I-projective resolution and take homology. +In particular, this yields extension groups Extn +I(A, B) for all A, B ∈∈ KK(X). Un- +like in usual homological algebra, Ext0 +I(A, B) may differ from the morphism space +in KK(X), compare the exact sequence (4.8) in [6]. + +4.2. Enough projective objects. A strategy to find enough projective objects is +outlined in [9, §3.6]. The idea is to study the left adjoint functor FK⊢ +Y of FKY ; this +is defined on P ∈∈ AbZ/2 if there is FK⊢ +Y (P) ∈∈ KK(X) and a natural isomorphism + +(4.3) +Hom +� +P, FKY (B) +� ∼= KK(X; FK⊢ +Y (P), B) + +for all B ∈∈ KK(X). Notice that FK⊢ +Y need not be defined for all P. +Objects of the form FK⊢ +Y (P) are automatically I-projective because the functor +KK(X; FK⊢ +Y (P), ) factors through FKY by (4.3) and vanishes on I by (4.2). +The simplest case to look for FK⊢ +Y (P) is P = Z[0] (this means Z in degree 0). +The defining property of FK⊢ +Y (Z[0]) is a natural isomorphism + +KK(X; FK⊢ +Y (Z[0]), B) ∼= Hom +� +Z[0], FKY (B) +� ∼= FKY,0(B) = K0 +� +B(Y ) +� +. + +In other words, FK⊢ +Y (Z[0]) must represent the covariant functor FKY . Theorem 2.5 +provides such representing objects, and yields the following: + +Proposition 4.1. For any Y ∈ LC(X), the adjoint functor FK⊢ +Y is defined on a +Z/2-graded Abelian group G = G0 ⊕ G1 if G0 and G1 are free and countable. More +precisely, + +FK⊢ +Y + +�� + +i∈I +Z[εi] + +� + += +� + +i∈I +RY [εi], + +where I is a countable set and εi ∈ Z/2 for all i ∈ I. + +Proof. We have just observed that FK⊢ +Y (Z[0]) = RY . Since FKY is stable, this +implies FK⊢ +Y (Z[1]) = RY [1]. It is a general feature of left adjoint functors that they +commute with direct sums. Since countable direct sums exist in KK(X), we get the +existence of FK⊢ +Y on any free countable Z/2-graded Abelian group. +□ + + +20 +RALF MEYER AND RYSZARD NEST + +Corollary 4.2. There are enough I-projective objects in KK(X), and the class of +I-projective objects in KK(X) is generated by the objects RY for Y ∈ LC(X)∗. +More precisely, any I-projective objects is a retract of a direct sum of suspensions +of these objects. + +Proof. This follows from Proposition 4.1 and [9, Proposition 3.37]. +□ + +Often we do not need retracts here, that is, any I-projective object is a direct +sum of suspensions of RY for Y ∈ LC(X)∗; for the totally ordered spaces studied +in §3, this follows from Theorem 3.12. +Since our ideal I is compatible with countable direct sums, the I-contractible +objects form a localising subcategory of KK(X), that is, they form a class NI of +objects that is closed under countable direct sums, retracts, isomorphism, exact +triangles, and suspensions. Furthermore, NI is the complement of the localising +subcategory that is generated by the I-projective objects. These two subcategories +contain much less information than the ideal itself. Roughly speaking, they will be +the same for any reasonable choice of invariant on KK(X) of K-theoretic nature. + +Proposition 4.3. The localising subcategory that is generated by the I-projective +objects is the bootstrap category B(X). It consists of all objects of KK(X) that are +KK(X)-equivalent to a tight, nuclear, purely infinite, stable, separable C∗-algebra +over X whose simple subquotients belong to the bootstrap category B ⊆ KK. + +Proof. By definition, B(X) is the localising subcategory of KK(X) that is generated +by the objects ix(C) for x ∈ X, see [8]. These generators are I-projective because +they represent the functors FKUx, compare the proof of the Representability The- +orem 2.5. The proof of this theorem also shows that the representing objects RY +belong to the triangulated subcategory of KK(X) generated by RUx for x ∈ X and +hence to B(X). Now Corollary 4.2 shows that all I-projective objects belong to +B(X). Hence the localising subcategory they generate is contained in the bootstrap +class. +Conversely, since the generators of the bootstrap class ix(C) are I-projective, +the localising subcategory generated by the I-projective objects must contain the +whole bootstrap class. This yields the first statement. The second one is contained +in [8, Corollary 5.5]. +□ + +4.3. The universality of filtrated K-theory. The next step in the general pro- +gramme is to determine the universal defining functor for I. This functor is char- +acterised by the universal property that it is I-exact and stable homological and +that any I-exact homological functor on KK(X) factors through it uniquely (up to +natural isomorphism). +The advantage of using the universal functor is that it describes I-projective +resolutions and the associated I-derived functors in KK(X) by projective resolutions +and derived functors in its target Abelian category. +This is the crucial step to +compute these derived functors. +In the presence of enough projective objects, [9, Theorem 3.39] characterises the +universal functor by an adjointness property. In our case, this yields: + +Theorem 4.4. The filtrated K-theory functor FK: KK(X) → Mod(NT )c is the +universal I-exact stable homological functor; here Mod(NT )c denotes the category +of all countable graded NT -modules. + +The ring of natural transformations NT comes in automatically at this point. + +Proof. This is best explained as a special case of a general result on certain homo- +logical ideals. Let T be any triangulated category with countable direct sums, and + + +C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY +21 + +let G be an at most countable set of objects of T. Let IG be the stable homological +ideal defined by the functor + +FG : T → +� + +G∈G +AbZ, +A �→ +� +T∗(G, A) +� + +G∈G. + +We assume that FG(A) is countable for all A ∈∈ T. +We are dealing with the case where T = KK(X) and G = {RY | Y ∈ LC(X)∗}; +Theorem 2.5 identifies T∗(RY , A) = KK∗(X; RY , A) ∼= K∗ +� +A(Y ) +� += FKY (A) for +all Y ∈ LC(X)∗, so that IG = I with I as in (4.1). +Viewing G as a full subcategory of T, it becomes a Z-graded pre-additive cat- +egory, so that we get a corresponding category Mod(Gop)c of countable graded right +modules. We can enrich the functor FG to a functor + +F ′ +G : T → Mod(Gop)c + +because the composition in T provides maps + +T∗(G′, A) ⊗ T∗(G, G′) → T∗(G, A) + +for all G, G′ ∈ G, A ∈∈ T, which form a right G-module structure on +� +T∗(G, A) +� + +G∈G. +We claim that the functor F ′ +G is the universal IG-exact functor. +In the case at hand, our description of the natural transformations FKY ⇒ FKZ +in §2.1 means that Mod(Gop)c = Mod(NT )c and F ′ +G = FK is filtrated K-theory +as defined in Definition 2.4. Hence it suffices to establish the claim above to finish +the proof of Theorem 4.4. +To do this, we check the conditions in [9, Theorem 3.39]. Idempotent morphisms +in KK(X) split because this happens in any triangulated category with countable +direct sums (see [10]). +Call F ′ +G(G) = T( , G) for G ∈ G the free Gop-module +on G. Direct sums of free modules are projective, and any object of Mod(Gop)c is a +quotient of a countable direct sum of free modules. Hence Mod(Gop)c has enough +projective objects. Moreover, + +HomGop� +F ′ +G(G), F ′ +G(A) +� ∼= F ′ +G(A)(G) = T(G, A) + +shows that the left adjoint F ⊢ of F := F ′ +G maps F ′ +G(G) to G ∈∈ T. Since the +domain of F ⊢ is closed under suspensions, countable direct sums, and retracts, the +adjoint is defined on all projective modules. Furthermore, F ◦ F ⊢(P) ∼= P holds +for free modules and hence for all projective modules P. Having checked all the +hypotheses of [9, Theorem 3.39], we can conclude that F ′ +G is indeed universal. +□ + +Since FK: KK(X) → Mod(NT )c is universal, [9, Theorem 3.41] now tells us, +roughly speaking, that homological algebra in KK(X) with respect to I is equivalent +to homological algebra in the Abelian category Mod(NT )c: +• An object A of KK(X) is I-projective if and only if FK(A) ∈ Mod(NT )c +is projective and + +KK∗(X; A, B) ∼= HomNT +� +FK(A), FK(B) +� + +for all B ∈∈ KK(X). +Another equivalent condition is that FK(A) ∈ Mod(NT )c is projective +and A belongs to the localising subcategory generated by the I-projective +objects; the latter agrees with the bootstrap class by Proposition 4.3. +• The functor FK and its partially defined left adjoint FK⊢ restrict to an +equivalence of categories between the subcategories of I-projective objects +in KK(X) and of projective objects in Mod(NT )c. +• For any A ∈∈ KK(X), the functors FK and FK⊢ induce bijections between +isomorphism classes of I-projective resolutions of A and isomorphism classes + + +22 +RALF MEYER AND RYSZARD NEST + +of projective resolutions of FK(A) in Mod(NT )c. That is, a projective res- +olution in Mod(NT )c lifts to a unique I-projective resolution in KK(X). +This provides the “geometric resolutions” that are used in connection with +the usual Universal Coefficient Theorem for KK. +• For all n ∈ N, there is a natural isomorphism + +Extn +I(A, B) ∼= Extn +NT +� +FK(A), FK(B) +� +, + +where the right hand side denotes extension groups in the Abelian category +Mod(NT )c. +• For any homological functor G: KK(X) → C, there is a unique right-exact +functor ¯G: Mod(NT )c → C with ¯G ◦ FK(P) = G(P) for all I-projective P. +The left derived functors of G with respect to I are Ln ¯G ◦ FK for n ∈ N, +where Ln ¯G: Mod(NT )c → C denotes the nth left derived functor of ¯G. + +4.4. The Universal Coefficient Theorem. In the general theory, the next step +is to construct a spectral sequence whose E2-term involves the extension groups +Extn +I(A[m], B); it converges—in favourable cases—to KK∗(X; A, B). This spectral +sequence is constructed in [3, 6]. Since we aim for an exact sequence, not for a +spectral sequence, we only need the special case considered in [9, Theorem 4.4]. +This provides the Universal Coefficient Theorem we want under the assumption +that FK(A) has a projective resolution of length 1 in Mod(NT )c: + +Theorem 4.5. Let A, B ∈∈ KK(X). Suppose that FK(A) ∈∈ Mod(NT )c has a +projective resolution of length 1 and that A ∈∈ B(X). Then there are natural short +exact sequences + +Ext1 +NT +� +FK(A)[j + 1], FK(B) +� +֌ KKj(X; A, B) ։ HomNT +� +FK(A)[j], FK(B) +� + +for j ∈ Z/2, where HomNT and Ext1 +NT denote the morphism and extension groups +in the Abelian category Mod(NT )c and [j] and [j + 1] denote degree shifts. + +The bootstrap class appears here because of Proposition 4.3, which identifies it +with the localising subcategory generated by the I-projective objects. + +Corollary 4.6. Let A, B ∈∈ B(X) and suppose that both FK(A) and FK(B) have +projective resolutions of length 1 in Mod(NT )c. Then any morphism FK(A) → +FK(B) in Mod(NT )c lifts to an element in KK0(X; A, B), and an isomorphism +FK(A) ∼= FK(B) lifts to an isomorphism in B(X). + +Proof. The lifting of a homomorphism follows from Theorem 4.5. Given an iso- +morphism f : FK(A) → FK(B), we can lift f and f −1 to elements α and β of +KK0(X; A, B) and KK0(X; B, A), respectively. Since β ◦ α lifts the identity map +on FK(A), the difference id − β ◦ α belongs to Ext1 +NT +� +FK(A)[j + 1], FK(A) +� +. The +latter is a nilpotent ideal in KK(X; A, A) because of the naturality of the exact +sequence in Theorem 4.5. Hence (id − βα)2 = 0, so that β ◦ α is invertible. The +same argument shows that α ◦ β is invertible, so that α is invertible. +□ + +This corollary is what is needed for the classification programme, and it depends +on resolutions having length 1. Conversely, if there is A for which FK(A) has no +projective resolution of length 1, then it is likely that there exist non-isomorphic +B, D ∈∈ B(X) with FK(B) ∼= FK(D). +The following theorem provides such a +counterexample, but under a stronger assumption. + +Theorem 4.7. Let I be a homological ideal in a triangulated category T with enough +I-projective objects. Let F : T → AIT be a universal I-exact stable homological +functor. Suppose that I2 ̸= 0. Then there exist non-isomorphic objects B, D ∈∈ T +for which F(B) ∼= F(D) in AIT. + + +C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY +23 + +Proof. Since I2 ̸= 0, there is A ∈∈ T with I2(A, ) ̸= 0, that is, A is not +I2-projective. The ideal I2 has enough projective objects as well, so that there +is an exact triangle +ΣN2 +γ2 +−→ ˜A2 +α2 +−→ A +ι2 +−→ N2 +with ι2 ∈ I2 and an I2-projective object ˜A2 (this is part of the phantom castle +constructed in [6], where the same notation is used). +Since ι2 ∈ I, this triangle is I-exact and hence provides an extension + +F(N2)[1] ֌ F( ˜A2) ։ F(A) + +in AIT. Even more, this extension splits because ι2 ∈ I2. This follows because the +canonical map +I(A, N2) → Ext1 +I(A, N2[1]) +implicitly used above factors through I/I2 and hence annihilates ι2 (see [6, Equa- +tion (4.9)]). As a result, F( ˜A2) ∼= F(A) ⊕ F(N2)[1]. +But ˜A2 cannot be isomorphic to A ⊕ N2[1]. If this were the case, then A would +be I2-projective, as a retract of the I2-projective object ˜A2. Then I2(A, ) = 0, +contradicting our choice of A. Hence ˜A2 ̸∼= A ⊕ N2[1]. +□ + +If I2 = 0, then the ABC spectral sequence constructed in [6] degenerates at the +third stage, that is, E3 = E∞. But E2 and E3 differ unless projective resolutions +have length 1. Hence the vanishing of I2 is probably not sufficient for isomorphisms +on the invariant to lift because the boundary map d2 on the second stage of the +ABC spectral sequence may provide further obstructions. +Whether or not filtrated K-theory gives rise to projective resolutions of length 1 +depends on the space in question: we will find positive and negative cases below. +Before we turn to examples, we discuss another important issue: does filtrated +K-theory exhaust all of Mod(NT )c? +This is definitely not the case because of +the additional exactness conditions that hold for objects of the form FK(A). The +following result is not optimal but sufficient for our purposes. + +Theorem 4.8. Let G ∈∈ Mod(NT )c have a projective resolution of length 1. Then +there is A ∈∈ B(X) with FK(A) ∼= G, and this object is unique up to isomorphism +in B(X). + +Proof. Any projective resolution of length 1 in Mod(NT )c is isomorphic to one of +the form +· · · → 0 → FK(P1) +FK(f) +−−−−→ FK(P0) → G +for suitable I-projective objects P1, P0 ∈∈ KK(X) and some f ∈ KK0(X; P1, P0). +Here we use that FK restricts to an equivalence of categories between the subcat- +egories of I-projective objects of KK(X) and of projective objects of Mod(NT )c by +the first paragraph of [9, Theorem 3.41]. +We may embed the morphism f in an exact triangle + +ΣA +h−→ P1 +f−→ P0 +g−→ A. + +Since FK(f) is injective, the map f is I-monic; thus g is I-epic and h ∈ I. Therefore, +the long exact sequence for FK applied to the above triangle degenerates to a short +exact sequence +FK(P1) ֌ FK(P0) ։ FK(A). +This yields FK(A) ∼= G as desired. The uniqueness of A is already contained in +Corollary 4.6. +□ + +It remains to understand which objects of the category Mod(NT )c have a pro- +jective resolution of length 1. + + +24 +RALF MEYER AND RYSZARD NEST + +4.5. Resolutions of length 1 in the totally ordered case. We return to the +example of the space X = {1, . . ., n} totally ordered by ≤ studied in §3. Let NT be +the graded pre-additive category of natural transformations described in §3, and let +C = Mod(NT )c be the Abelian category of NT -modules. The following theorem +characterises NT -modules with projective resolutions of length 1: + +Theorem 4.9. Let M ∈∈ C. The following assertions are equivalent: +(i) M = FK∗(A) for some A ∈∈ KK(X); +(ii) M is exact in the sense of Definition 3.5; +(iii) TorNT +i +(NT ss, M) = 0 for i = 1, 2; +(iv) M has a free resolution of length 1 in C; +(v) M has a projective resolution of length 1 in C; +(vi) M has a projective resolution of finite length in C. + +Proof. The exact sequence (1.4) shows that (i) implies (ii). Theorem 4.8 contains +the implication (v)=⇒(i), and the implications (iv)=⇒(v)=⇒(vi) are trivial. We +will show (ii)=⇒(iii)=⇒(iv) and (vi)=⇒(ii), and this will establish the theorem. +First we show that (vi) implies (ii). +Let 0 → Pm → · · · → P0 → M be a +projective resolution of finite length. By a standard “stabilisation” trick, we can +turn this into a free resolution of the same length. Let + +Zj = ker(Pj → Pj−1) ∼= range(Pj+1 → Pj). + +Thus Zm = 0, P0/Z0 ∼= M, and we have exact sequences Zj ֌ Pj ։ Zj−1 +because our chain complex is exact. Since Zm = 0, the exactness of the projective +modules Pm and Proposition 3.6 show recursively that Zj is exact for j = m − +1, m − 2, . . . , 0, so that M is exact. Thus (vi) implies (ii). +Now we prove (ii)=⇒(iii)=⇒(iv). Let P be a countable free module for which +there is an epimorphism π: P ։ M, and let K := ker π. We have an extension +of NT -modules K ֌ P ։ M. Proposition 3.6 shows that K is exact because P +and M are exact. Furthermore, Tori+1(NT ss, M) ∼= Tori(NT ss, K) for all i ≥ 1 +because P is projective. Lemma 3.13 applied to M and K yields Tori(NT ss, M) = 0 +for i = 1, 2 if M is exact, that is, (ii)=⇒(iii). Now assume (iii). The argument above +yields Tor1(NT ss, K) = 0. Since P is projective, the Abelian groups P(Y ) are free +for all Y ∈ LC(X). The exact sequence in (3.5) yields the same for K(Y ). The +criterion in Theorem 3.12.(3) shows that K is projective. +□ + +Now we combine the existence of projective resolutions of length 1 with The- +orem 4.5, which still required this as a hypothesis: + +Theorem 4.10. Let X be the topological space associated to a totally ordered finite +set, and let A and B be C∗-algebras over X. If A ∈∈ B(X), then there is a natural +short exact sequence + +Ext1 +NT +� +FK(A)[1], FK(B) +� +֌ KK∗(X; A, B) ։ HomNT +� +FK(A), FK(B) +� +. + +In particular, any NT -module morphism FK(A) → FK(B) lifts to an element +in KK∗(X; A, B). If both A and B belong to the bootstrap class B(X), then an +isomorphism FK(A) ∼= FK(B) lifts to a KK-equivalence A ≃ B. + +Proof. Use Theorem 4.5 and Corollary 4.6 together with the existence of projective +resolutions of length 1 ensured by Theorem 4.9. +□ + +Theorem 4.11. Let X be the topological space associated to a totally ordered finite +set, and let A and B be tight, purely infinite, stable, nuclear, separable C∗-algebras +over X whose simple subquotients belong to the bootstrap category. Then an iso- +morphism FK(A) ∼= FK(B) lifts to an X-equivariant ∗-isomorphism A ∼= B. + + +C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY +25 + +Furthermore, any countable exact NT -modules is the filtrated K-module of some +tight, purely infinite, stable, nuclear, separable C∗-algebra over X with simple sub- +quotients in the bootstrap category. + +Proof. A nuclear C∗-algebras over X belongs to the bootstrap category B(X) if and +only if its fibres belong to the non-equivariant bootstrap category B (see [8, Corol- +lary 4.13]). For a tight C∗-algebra over X, these fibres are the same as the simple +subquotients. +It is also shown in [8, Corollary 5.5] that any object of B(X) is +KK(X)-equivalent to a tight, nuclear, purely infinite, simple, separable C∗-algebra +over X whose simple subquotients belong to the bootstrap category B. A deep clas- +sification result of Eberhard Kirchberg shows that any KK(X)-equivalence between +such objects lifts to an X-equivariant ∗-homomorphism. Now the first assertion fol- +lows from Theorem 4.10. The second assertion also uses Theorem 4.8. +□ + +5. A counterexample + +Now we let X := {1, 2, 3, 4} with the partial order 1, 2, 3 < 4 and no relation +among 1, 2, 3. Hence the open subsets of X are + +O(X) = +� +∅, {4}, {1, 4}, {2, 4}, {3, 4}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}, {1, 2, 3, 4} +� +, + +that is, a non-empty subset is open if and only if it contains 4. The associated +directed graph is +• 1 + +4 • +� +❑❑ �❑ +❑ + +sss �s +• 2 + +• 3. + +We frequently denote subsets of X simply by 124 := {1, 2, 4}, and so on. +A C∗-algebra over X is a C∗-algebra A with four distinguished ideals + +I1 := A(14), +I2 := A(24), +I3 := A(34), +I4 := A(4), + +such that I1 + I2 + I3 = A and Ii ∩ Ij = I4 for all 1 ≤ i < j ≤ 3 (see [8, Lemma +2.35]). Equivalently, the ideals Ij/I4 for j = 1, 2, 3 decompose A/I4 into a direct +sum of three orthogonal ideals. The other distinguished ideals are + +A(124) = I1 + I2, +A(134) = I1 + I3, +A(234) = I2 + I3. + +Any subset of X is locally closed. But a connected locally closed subset is either +open or one of the singletons {1}, {2}, and {3}. Hence the set of connected locally +closed subsets is + +LC(X)∗ = {4, 14, 24, 34, 124, 134, 234, 1234, 1, 2, 3}. + +The order complex Ch(X) is a graph with four vertices 1, 2, 3, 4 and edges joining +the first three to the last one: + +Ch(X) = + +���� +���� +1 +▲▲▲▲▲ +▲ + +���� +���� +2 +���� +���� +4 + +���� +���� +3 + +rrrrr +r + +Both maps m, M : Ch(X) → X map the vertices to the corresponding points in X. +Whereas M maps the interior of each edge to 4, the map m maps the interior of +the edge [j, 4] to j for j = 1, 2, 3. +Recall that the space of natural transformations FKY ⇒ FKZ is given by + +NT ∗(Y, Z) ∼= K∗� +S(Y, Z) +� +, +S(Y, Z) := m−1(Y ) ∩ M −1(Z) ⊆ Ch(X). + + +26 +RALF MEYER AND RYSZARD NEST + +Y \Z +4 +14 +24 +34 +124 +134 +234 +1234 +1 +2 +3 + +4 +Z +Z +Z +Z +Z +Z +Z +Z +0 +0 +0 + +14 +0 +Z +0 +0 +Z +Z +0 +Z +Z +0 +0 + +24 +0 +0 +Z +0 +Z +0 +Z +Z +0 +Z +0 + +34 +0 +0 +0 +Z +0 +Z +Z +Z +0 +0 +Z + +124 +Z[1] +0 +0 +Z[1] +Z +0 +0 +Z +Z +Z +0 + +134 +Z[1] +0 +Z[1] +0 +0 +Z +0 +Z +Z +0 +Z + +234 +Z[1] +Z[1] +0 +0 +0 +0 +Z +Z +0 +Z +Z + +1234 +Z[1]2 +Z[1] +Z[1] +Z[1] +0 +0 +0 +Z +Z +Z +Z + +1 +Z[1] +0 +Z[1] +Z[1] +0 +0 +Z[1] +0 +Z +0 +0 + +2 +Z[1] +Z[1] +0 +Z[1] +0 +Z[1] +0 +0 +0 +Z +0 + +3 +Z[1] +Z[1] +Z[1] +0 +Z[1] +0 +0 +0 +0 +0 +Z + +Table 1. The ring of natural transformations + +It is straightforward to compute these K-theory groups, and the results are listed +in Table 1. +Here the rows are labelled by Y , the columns by Z. +For instance, +the entry Z at (14, 1) means that NT ∗(14, 1) ∼= Z. +The trivial 1-dimensional +bundle over S(14, 1) generates this group. +Hence Remark 2.12 shows that the +generator is the natural transformation that we get from the quotient map A(14) ։ +A(1). +Similar arguments show that all the natural transformations of degree 0 +are induced by the familiar restriction and extension ∗-homomorphisms for closed +and open subsets. Moreover, the odd natural transformations arise by composing +these ∗-homomorphisms with boundary maps in K-theory long exact sequences. All +relations that they satisfy are predicted by morphisms of extensions and exactness +of the sequences (1.4). +The computations in §3 were based on a description of indecomposable morph- +isms in the category NT ∗. For the space X in question, these are the maps in the +following diagram: + +(5.1) + +14 +i +� + +i +■■■■■■■■ �■ +124 + +i +❑❑❑❑❑❑❑❑ �❑ +1 + +◦❊ +❊❊ +❊ + +δ + +❊❊ �❊ +❊ + +4 + +i +①①①①①①① �① +① +i +� + +i + +❋❋❋❋❋❋❋❋ �❋ +24 + +i +✉✉✉✉✉✉✉ �✉ +✉ + +i +■■■■■■■■ �■ +134 +i +� 1234 + +r +✉✉✉✉✉✉✉✉ �✉ +r +� + +r + +■■■■■■■■■ �■ +2 +◦δ +� 4 + +34 +i +� + +i +✉✉✉✉✉✉✉✉ �✉ +234 + +i +sssssssss �s +3 + +②② ◦② +② + +δ +②② �② +② + +Here we write i for the extension transformation for an open subset, r for the +restriction transformation for a closed subset, and δ for boundary maps in K-theory +long exact sequences. +The indecomposable morphisms in (5.1) provide a minimal set of generators for +the graded ring NT . To describe NT completely, we list the relations. These are +generated by the following: +• the cube with vertices 4, 14, . . . , 1234 is a commuting diagram, that is, all +the commuting squares involving arrows with label i commute; +• the following composite arrows vanish: + +124 +i−→ 1234 +r−→ 3, +134 +i−→ 1234 +r−→ 2, +234 +i−→ 1234 +r−→ 1, + +1 +δ−→ 4 +i−→ 14, +2 +δ−→ 4 +i−→ 24, +3 +δ−→ 4 +i−→ 34; + +• the sum of the three maps 1234 → 4 via 1, 2, and 3 vanishes. + + +C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY +27 + +These relations imply that the diagrams + +124 +r +� + +r +� + +− + +2 + +◦δ +� + +1 +◦δ +� 4 + +134 +r +� + +r +� + +− + +3 + +◦δ +� + +1 +◦δ +� 4 + +234 +r +� + +r +� + +− + +2 + +◦δ +� + +3 +◦δ +� 4 + +anti-commute and that the composite of two odd maps vanishes. It is routine to +check that the universal pre-additive category with these generators and relations +is given by the groups listed in Table 1. +Define NT nil and NT ss as in Definition 3.3: NT nil is the linear span of the +groups NT ∗(Y, Z) with Y ̸= Z and NT ss is spanned by the groups NT ∗(Y, Y ). +Then NT nil is a nilpotent ideal in NT and NT ss ∼= ZLC(X)∗ is a semi-simple +ring. Thus NT nil is the maximal nilpotent ideal in NT and we have a semi-direct +product decomposition NT ∼= NT nil ⋊ NT ss as in Lemma 3.4. +The next task is to describe the submodule M ′ := NT nil · M ⊆ M for an exact +NT -module M. The following computations are done as in the proof of Lemma 3.8, +using (3.4) and that the morphisms in (5.1) generate NT . + +M ′(14) = range +� +i14 +4 : M(4) → M(14) +� += ker +� +r1 +14 : M(14) → M(1) +� +, + +and symmetrically for 24 and 34; + +M ′(124) = range +� +i124 +14 : M(14) → M(124) +� ++ +� +i124 +24 : M(24) → M(124) +� + += ker +� +δ4 +124 : M(124) → M(4) +� +, + +where δ4 +124 denotes a generator of NT 1(124, 4) ∼= Z; symmetry provides M ′(134) +and M ′(234). We have + +M ′(1) = range +� +r1 +1234 : M(1234) → M(1) +� += ker +� +δ234 +1 +: M(1) → M(234) +� +, + +and symmetrically for 2 and 3, and + +M ′(4) = + +3 +� + +j=1 +range +� +δ4 +j : M(j) → M(4) +� += ker +� +i1234 +4 +: M(4) → M(1234) +� +. + +But something goes wrong with M ′(1234). Equation (3.4) yields + +range +� +i1234 +124 : M(124) → M(1234) +� ++ +� +i1234 +134 : M(134) → M(1234) +� + += ker +� +δ14 +1234 : M(1234) → M(14) +� +; + +to take into account the range of i1234 +234 as well, we need an exact sequence containing +δ14 +1234 ◦ i1234 +234 , which is the generator of NT 1(234, 14) ∼= Z. Since there is no such +exact sequence, our method breaks down at this point. +Another symptom but not a cause of problems is that the map δ4 +124 that describes +M ′(124) is not the longest map out of 124: that would be δ34 +124. +As we shall see, the analogues of Theorems 3.12 and 4.9 become false for the +space X. First, there is a non-projective exact module M with free Mss; secondly, +there is a module that has no projective resolution of length 1; thirdly, there are +A, B ∈ B(X) with I2(A, B) ̸= 0. +Hence Theorem 4.7 provides non-isomorphic +objects in the bootstrap class B(X) with isomorphic filtrated K-theory. The con- +struction of these counterexamples follows the above pattern: first we find a counter- +example to Theorem 3.12, which we use to find one for Theorem 4.9, which is then +used to find an example as in Theorem 4.7. +We begin with the unexpected non-projective module. Let PY for Y ∈ LC(X)∗ + +denote the free NT -module on Y , that is, + +PY (Z) = NT ∗(Y, Z), +HomNT (PY , N) ∼= N(Y ) + + +28 +RALF MEYER AND RYSZARD NEST + +for any Y, Z ∈ LC(X)∗ and any NT -module N. A natural transformation FKY ⇒ +FKZ corresponds to an element in NT ∗(Y, Z) ∼= PY (Z) ∼= HomNT (Pz, PY ) and +thus induces a module homomorphism PZ → PY in the opposite direction. Hence +the three arrows 124, 134, 234 → 1234 in (5.1) induce a module homomorphism + +j : P1234 → P 0 := P124 ⊕ P134 ⊕ P234. + +Table 1 shows that there are no module homomorphisms P 0 → P1234, that is, no +non-zero natural transformations from 1234 to 124, 134, or 234. +The crucial observation is that j is a monomorphism, so that P1234 becomes a +submodule of P 0. Since the longest natural transformations out of 1234 are those +to 14, 24 and 34, this follows from the elementary observations that the maps + +NT ∗(1234, j4) → NT ∗(1234 \ j, j4) + +for j = 1, 2, 3 are, respectively, the identity map on Z. +This follows from the +exactness of free modules because NT ∗(j, j4) = 0 by Table 1. +We describe the quotient + +M := P 0/j(P1234) + +by its values M(Y ) for Y ∈ LC(X)∗ as in (5.1): + +(5.2) + +0 +i +� + +i + +❉❉❉❉❉❉❉❉ �❉ +Z + +i + +❋❋❋❋❋❋❋❋ �❋ +Z + +◦● +●●● +●●● +● + +δ +●●●● +●●● +● + +Z[1] + +i +✇✇✇✇✇✇✇ �✇ +✇ +i +� + +i + +●●●●●●●● �● +0 + +i +③③③③③③③ �③ +③ + +i + +❉❉❉❉❉❉❉❉ �❉ +Z +i +� Z2 + +r +①①①①①①① �① +① +r +� + +r + +❋❋ �❋ +❋ +❋ +❋❋ +❋ +❋ +❋ +Z +◦δ +Z[1] + +0 +i +� + +i +③③③③③③③③ �③ +Z + +i +① �① +① +① +① +①① +① +① +① +Z + +✇✇✇◦ +✇ ✇ +✇✇ +✇ + +δ +✇✇✇ +✇ ✇ +✇✇ +✇ + +The boundary maps δ act by isomorphisms on M because M(j4) = 0 for j = +1, 2, 3. The other maps can be understood by writing M(1234) = Z3/⟨(1, 1, 1)⟩ and +M(j) = Z2/⟨(1, 1)⟩ for j = 1, 2, 3 as quotients. The three maps Z → Z2 correspond +to the three coordinate embeddings Z ֌ Z3, the maps Z2 → Z to the projections +Z3 ։ Z2 onto coordinate hyperplanes. +The projective resolution + +(5.3) +0 → P1234 +j−→ P 0 ։ M + +does not split because there exist no non-zero morphisms P 0 → P1234. Hence M +is not projective. But Mss is free, and M is exact because the exact modules form +an exact category and P1234 and P 0 are exact. Thus M is a counterexample to +Theorem 3.12. +The module M is directly related to the problem with describing NT nil·M(1234) +encountered above. Since HomNT (PY , N) ∼= N(Y ) for any NT -module N and any +Y ∈ LC(X)∗, the resolution (5.3) provides an exact sequence + +0 → HomNT (M, N) + +→ N(124) ⊕ N(134) ⊕ N(234) → N(1234) → Ext1 +NT (M, N) → 0, + +so that +Ext1 +NT (M, N) ∼= N(1234)/NT nil · N(1234) ∼= Nss(1234). +Now we use M to construct a counterexample for Theorem 4.9. Let k ∈ N≥2 +and let Mk := M/k · M; that is, we replace Z by Z/k everywhere in (5.2). This +module has a projective resolution of length 2 of the form + +(5.4) +0 → P1234 +(−k,j) +−−−−→ P1234 ⊕ P 0 +(j,k) +−−−→ P 0 ։ Mk, + + +C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY +29 + +where k denotes multiplication by k. Using this resolution, we compute + +Ext2(Mk, P1234) ∼= Z/k, +Ext1(Mk, P1234) ∼= Hom(Mk, P1234) ∼= 0 + +because there are no no-zero morphisms P 0 → P1234. Of course, the generator of +Ext2(Mk, P1234) is the class of the projective resolution (5.4). Hence Mk admits no +projective resolution of length 1 and is a counterexample to Theorem 4.9. +Now we claim that Mk is the filtrated K-theory of some C∗-algebra Ak over X in +the bootstrap class B(X). To begin with, M is the filtrated K-theory of some such +C∗-algebra A by Theorem 4.8. Let Bk be a C∗-algebra in the bootstrap class with +K0(Bk) = Z/k and K1(Bk) = 0; for instance, Bk could be the Cuntz algebra Ok+1. +Then Ak := A ⊗ Bk has filtrated K-theory Mk by the K¨unneth Theorem for the +K-theory of tensor products. + +Theorem 5.1. Let Ak be a C∗-algebra in the bootstrap class with FK(Ak) ∼= Mk +as constructed above. Then Ak is not I2-projective. Hence there exist B, D ∈ B(X) +that are not KK(X)-equivalent but with the same filtrated K-theory. + +Proof. The second assertion follows from the first one using Theorem 4.7 applied +to the bootstrap class B(X) and the restriction of I to B(X). +It remains to prove that Ak cannot be I2-projective. To see this, we lift the +resolution (5.4) to an I-projective resolution + +0 +◦ +�P2 +◦ +�P1 +◦ +�P0 +�Ak + +in B(X) with boundary maps of degree 1, and embed the latter in a phantom tower +(see [6]): + +Ak +N0 +ι1 +0 +� N1 +ι2 +1 +� + +◦✠✠✠ + +�✠✠✠ + +N2 +ι3 +2 +� + +◦✠✠✠ + +�✠✠✠ + +N3 + +◦✠✠✠ + +�✠✠✠ + +N3 + +◦☛☛☛ + +�☛☛☛ + +· · · + +P0 + +π0 + +�✺✺✺✺✺✺ +P1 + +π1 + +�✺✺✺✺✺✺ +� +P2 + +π2 + +�✺✺✺✺✺✺ +� +0 + +�✸✸✸✸✸✸ +� +· · · +� + +The inductive system (Nj, ιj+1 +j +) becomes constant at N3 because Pj = 0 for j ≥ 3. +Since Ak belongs to the bootstrap class, N3 ∼= 0 (see the proof of [6, Proposition +4.5]). This implies N2 ∼= P2. +The composite map ι2 +0 : Ak = N0 → N2 ∼= P2 belongs to I2. Suppose that Ak +were I2-projective. Then ι2 +0 = ι2 +1 ◦ ι1 +0 would vanish, and the long exact homology +sequence would yield that the map ι2 +1 : N1 → N2 must factor through the map +N1 → P0. But + +KK∗(X; P0, P2) ∼= HomNT +� +FK(P0), FK(P2) +� += HomNT (P 0, P1234) = 0. + +Here we have used that filtrated K-theory, by universality, is fully faithful on +I-projective objects and that there are no non-zero module homomorphisms P 0 → +P1234. Since ι2 +1 factors through the zero group, it must be the zero map. But then +the map P1 → N1 must be a split surjection, so that N1 is I-projective. Then the +I-exact triangle ΣAk → ΣN1 → P0 → Ak provides an I-projective resolution of Ak +of length 1, which is impossible because FK(Ak) ∼= Mk has no projective resolution +of length 1. As a consequence, Ak is not I2-projective. +□ + +We can make the two non-equivalent C∗-algebras over X with the same filtrated +K-theory more explicit. One of them is Ak ⊕ ΣR1234, the other one is the mapping +cone of the map ι2 +0 : Ak = N0 → N2 ∼= R1234 in the phantom tower above. Both +have Mk ⊕ P1234[1] as their filtrated K-theory. +This counterexample shows that filtrated K-theory does not yet classify purely +infinite stable nuclear separable C∗-algebras in the bootstrap class. + + +30 +RALF MEYER AND RYSZARD NEST + +Remark 5.2. Refining filtrated K-theory by taking filtrated K-theory with coeffi- +cients does not help. This gets rid of the counterexample Ak constructed above, +but other objects of B(X) without projective resolution of length 1 remain. An ex- +ample is A ⊗ B, where B is a C∗-algebra in the bootstrap class with K∗(B) = Q[0] +such as an appropriate UHF-algebra. Its filtrated K-theory is M ⊗ Q. This also +has cohomological dimension 2, and this is not affected much by taking K-theory +with coefficients because M ⊗ Q is torsion-free. + +5.1. A refined invariant. There are at least two ways to identify the source of +the problem for the space X. The first point of view is that what is missing is an +exact sequence that has the generator α of NT 1(234, 14) as its connecting map. +The map α corresponds to a map ΣR14 → R234 between the representing objects, +which we also denote by α. In the triangulated category KK(X), we can embed the +latter map in an exact triangle + +(5.5) +ΣR14 +α−→ R234 → R12344 → R14. + +The notation R12344 will be explained later. The functors these objects represent +sit in a long exact sequence + +(5.6) +· · · → FK14 → FK12344 → FK234 +α−→ FK14[1] → · · · + +which is precisely what we want. The second point of view is that the troublemaker +is the non-projective module M. Since M has a projective resolution of length 1, +there is a unique object in the bootstrap class with filtrated K-theory M. Actually, +this yields the same object as the first point of view: + +Lemma 5.3. The non-projective module M above agrees with FK(R12344). + +Proof. The map FKY (α) vanishes for almost all Y ∈ LC(X)∗ simply because the +graded groups involved have different parity or one of them vanishes. The only +exception is Y = 14. The group FK14(R14) = NT (14, 14) is generated by the +identity natural transformation. Since α is the generator of NT 1(234, 14), the map +FK14(α) is invertible. +Now we apply FK to the long exact sequence for the given exact triangle. Since +FK(α) vanishes on most Y and is invertible for Y = 14, we can easily compute the +groups FKY (R12344). We get the same groups as for the module M. It remains to +check that the isomorphism can be chosen as an NT -module homomorphism. The +main step is to check that the map + +Z2 ∼= FK124(R12344) ⊕ FK134(R12344) → FK1234(R12344) ∼= Z2 + +is invertible. Together with the known relations between the various natural trans- +formations, this implies the assertion. We omit the details of this computation. +□ + +The representing object R12344 is an algebra of functions on a two-dimensional +simplicial complex, which we do not describe here because it is not illuminating. +The functor that it represents, however, can be described rather nicely as follows. +Let A be a C∗-algebra over X. Pull back the extension A(14) ֌ A(124) ։ A(2) +along the quotient map A(234) ։ A(2) to an extension A(14) ֌ A(12344) ։ +A(234). The object R12344 represents the functor + +(5.7) +KK∗(X; R12344, A) ∼= K∗ +� +A(12344) +� +. + +To see this, two observations are necessary. First, K∗ +� +R12344(12344) +� ∼= Z; the +generator of this group yields a natural transformation between the two functors +in (5.7). Secondly, this natural transformation is invertible. This follows from the +Five Lemma, once we know that it extends the known natural isomorphisms + +KK∗(X; RY , A) ∼= K∗ +� +A(Y ) +� + + +C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY +31 + +for Y = 14 and Y = 234 to a chain map between the long exact sequences that +we get from (5.5) and from the extension A(14) ֌ A(12344) ։ A(234). +This +extension also explains the notation R12344. +Now we augment filtrated K-theory by adding the covariant functor + +B �→ FK12344(B) := K∗ +� +A(12344) +� ∼= KK∗(X; R12344, B). + +The new invariant takes values in the category of countable NT ′-modules, where +NT ′ is the Z/2-graded category whose object set is LC′ := LC(X)∗ ⊔ {12344} and +whose morphisms are the natural transformations between the various filtrated +K-groups, including now also FK12344. These natural transformations can be com- +puted by the Yoneda Lemma: + +NT ′ +∗(Y, Z) ∼= KK∗(X; RZ, RY ) ∼= FKZ(RY ) + +holds for all Y, Z ∈ LC′. The category ring for NT ′ +∗ is simply the ring KK∗(X; R, R) +where +R := +� + +Y ∈LC′ +RY . + +We replace the ideal I in KK(X) studied above by the kernel I′ of the enriched +filtrated K-theory functor + +FK′ : KK(X) → Mod(NT ′)c. + +The same arguments as above show that there are enough I′-projective objects and +that FK′ is the universal I′-exact stable homological functor. +The passage from I to I′ has improved the situation because R12344 has now +been promoted to an I′-projective object and, therefore, ceases to cause trouble. +In principle, something similar can be done in great generality: whenever we have +an object of the Abelian approximation that has a projective resolution of length 1, +we can lift it uniquely to an object of the triangulated category and refine the +ideal by intersecting it with the kernel of the functor this lifted object represents. +However, the policy to quieten troublemakers by promotion has the tendency to +encourage new troublemakers, so that it is not clear whether this general strategy +always resolves all problems after finitely many steps. But in the relatively simple +example at hand, this turns out to be the case. +To check this, we must describe the category NT ′. If Y, Z ∈ LC(X)∗, then +NT ′ +∗(Y, Z) = NT ∗(Y, Z) is given by the table on page 26. Furthermore, if Z ∈ +LC(X)∗, then NT ′ +∗(12344, Z) ∼= FKZ(R12344) = M(Z) by Lemma 5.3, and this is +described in (5.2). The upshot is: +• there are even natural transformations from FK12344 to FK124, FK134, +FK234—the generators of the respective groups of natural transformations— +such that any natural transformation FK12344 ⇒ FKZ with Z ∈ LC(X)∗ + +is a sum of natural transformations that factor through one of these three +and a natural transformation FKij4 ⇒ FKZ; +• the sum of the three natural transformations FK12344 ⇒ FK1234 via FK124, +FK134 and FK234 vanishes, and all other relations follow from these and +the already known ones listed after (5.1). +The exact triangle (5.5) yields a long exact sequence + +· · · → NT ′ +∗+1(Y, 234) +α−→ NT ′ +∗(Y, 14) → NT ′ +∗(Y, 12344) → NT ′ +∗(Y, 234) → · · · , + +which we may use to compute NT ′ +∗(Y, 12344) for all Y ∈ LC′. The map α induces +an isomorphism for Y = 234 and the zero map for all other Y because the source +and target have opposite parity or one of them vanishes. Thus + +Y +4 +14, 24, 34 +124, 134, 234 +1234 +1, 2, 3 +12344 + +NT ′ +∗(Y, 12344) +Z2 +Z +0 +Z[1] +Z[1] +Z + + +32 +RALF MEYER AND RYSZARD NEST + +These groups inherit from M their invariance under permutations of 1, 2, 3. Inspect- +ing composition with natural transformations in NT , we arrive at the following: + +• there are even natural transformations FKj4 ⇒ FK12344 for j = 1, 2, 3, such +that any natural transformation FKY ⇒ FK12344 with Y ∈ LC(X)∗ factors +through one of them; +• the sum of the three natural transformations FK4 ⇒ FK12344 vanishes, +• the natural transformations FKj4 ⇒ FK1234\j via FK12344 vanish; +• all other relations follow from these and the already known ones. + +As one may expect, the basic natural transformations FK14 ⇒ FK12344 ⇒ FK234 +are induced by the maps R234 → R12344 → R14 in the exact triangle (5.5). +The indecomposable morphisms of the new category NT ′ are the maps in the +following diagram: + +14 + +❋❋❋❋❋❋❋ �❋ +124 + +❋❋❋❋❋❋❋ �❋ +1 + +◦❃❃❃❃ + +�❃❃❃❃ + +4 + +⑦ �⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +� + +❅❅ �❅ +❅ +❅❅ +❅ +24 +� 12344 +� + +✈✈✈✈✈✈✈ �✈ +✈ + +❍❍❍❍❍❍❍❍ �❍ +134 +� 1234 + +③③③ �③ +③③③③ +� + +❉ �❉ +❉ +❉ +❉ +❉ +❉ +❉ +❉ +2 +◦ +� 4 + +34 + +① �① +①① +①① +①① +① +234 + +①①①①①①① �① +3 + +◦���� + +����� + +The category ring of NT ′ again has the by now familiar structure: it is a split +nilpotent extension of the semisimple algebra NT ′ +ss ∼= ZLC′ spanned by the identity +transformations on the objects and a nilpotent ideal NT ′ +nil that is the subgroup +generated by NT ′(Y, Z) with Y ̸= Z. + +Definition 5.4. A module over NT ′ is exact if it is exact as an NT -module and +the three sequences + +· · · → N∗+1(ij4) → N∗(k4) → N∗(12344) → N∗(ij4) → · · · + +for {i, j, k} = {1, 2, 3} are exact as well. + +The range of the invariant FK′ consists of exact NT ′-modules; the three new +exact sequences are, in fact, equivalent for symmetry reasons, and the extension + +· · · → N∗+1(234) → N∗(14) → N∗(12344) → N∗(234) → · · · + +is built into the definition of FK12344. +Let N be an exact NT ′-module and let N ′ := NT ′ +nil · N. The description of +N ′(14), N ′(1), and N ′(4) is the same as for the category NT , so that these groups +remain kernels of certain maps, as needed. Furthermore, N ′(1234) is the kernel of +the map N(1234) → N(12344)[1] induced by the generator of NT 1(1234, 12344), +so that the problem that appeared for the category NT is cured. +The computation of N ′(124) changes because this group is now the range of the +arrow N(12344) → N(124). But this is part of a long exact sequence because N is +exact, and we get +N ′(124) = ker +� +N(124) → N(34)[1] +� +, + +and similarly for N ′(134) and N ′(234). +Finally, N ′(12344) is the sum of the ranges of the maps N(j4) → N(12344) for +j = 1, 2, 3. Using exactness, we identify this in two steps with the kernel of the +map N(12344) → N(4)[1] induced by the generator of NT ′ +1(12344, 4). +As a result, the submodule NT ′ +nil·N is described using kernels of maps N(Y ) → +N(Z). By the way, these arrows are the longest arrows starting at Y as in Re- +mark 3.9. The same arguments as for totally ordered spaces now show: + + +C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY +33 + +Theorem 5.5. An NT ′-module N is free if and only if it is projective, if and only +if it is exact and N(Y ) is a free group for all Y ∈ LC′. + +Theorem 5.6. An NT ′-module N has a projective resolution of length 1 if and +only if it is exact. + +Theorem 5.7. Let A and B be C∗-algebras over the four-point space X under +consideration. +If A belongs to the bootstrap class B(X), then there is a natural +short exact sequence + +Ext1 +NT ′ +� +FK′(A)[1], FK′(B) +� +֌ KK∗(X; A, B) ։ HomNT ′� +FK′(A), FK′(B) +� +. + +In particular, morphisms FK′(A) → FK′(B) lift to elements in KK∗(X; A, B). If +both A and B belong to the bootstrap class, then an isomorphism FK′(A) ∼= FK′(B) +lifts to a KK(X)-equivalence. + +Corollary 5.8. The map A �→ FK′(A) is a bijection between the set of isomorphism +classes of tight, stable, purely infinite, separable, nuclear C∗-algebras over X with +simple subquotients in the bootstrap class and the set of isomorphism classes of +countable exact NT ′-modules. + +6. Conclusion + +We have obtained a Universal Coefficient Theorem that computes KK∗(X; A, B) +for A in the bootstrap class and X of a very special form, namely, {1, . . . , n} with +the Alexandrov topology from the total order. This Universal Coefficient Theorem +can be used to carry over classification results for simple, nuclear, purely infinite +C∗-algebras to nuclear, purely infinite C∗-algebras with primitive ideal space X, +using filtrated K-theory as the invariant. +For general finite topological spaces X, we still get a spectral sequence that +computes KK∗(X; A, B) using filtrated K-theory, but this spectral sequence need +not degenerate to an exact sequence, so that isomorphisms on filtrated K-theory +need not lift to X-equivariant KK-equivalences. In fact, we have found a counter- +example. At the same time, we were able to fix the counterexample by refining +filtrated K-theory. It is unclear whether such a refinement is available for all finite +topological spaces and how it looks like. + +References + +[1] Apostolos Beligiannis, Relative homological algebra and purity in triangulated categories, J. +Algebra 227 (2000), no. 1, 268–361, doi: 10.1006/jabr.1999.8237. MR 1754234 +[2] Alexander +Bonkat, +Bivariante +K-Theorie +f¨ur +Kategorien +projektiver +Systeme +von +C∗-Algebren, +Ph.D. +Thesis, +Westf. +Wilhelms-Universit¨at +M¨unster, +2002, +http://deposit.ddb.de/cgi-bin/dokserv?idn=967387191 (German). +[3] J. Daniel Christensen, Ideals in triangulated categories: phantoms, ghosts and skeleta, Adv. +Math. 136 (1998), no. 2, 284–339, doi: 10.1006/aima.1998.1735. MR 1626856 +[4] Samuel Eilenberg and John Coleman Moore, Foundations of relative homological algebra, +Mem. Amer. Math. Soc. No. 55 (1965), 39. MR 0178036 +[5] Eberhard Kirchberg, Das nicht-kommutative Michael-Auswahlprinzip und die Klassifikation +nicht-einfacher Algebren, C∗-Algebras (M¨unster, 1999), Springer, Berlin, 2000, pp. 92–141 +(German). MR 1796912 +[6] Ralf Meyer, Homological algebra in bivariant K-theory and other triangulated categories. II, +Tbil. Math. J. 1 (2008), 165–210. MR 2563811 +[7] Ralf Meyer and Ryszard Nest, The Baum–Connes conjecture via localisation of categories, +Topology 45 (2006), no. 2, 209–259, doi: 10.1016/j.top.2005.07.001. MR 2193334 +[8] +, C∗-Algebras over topological spaces: the bootstrap class, M¨unster J. Math. 2 (2009), +215–252. MR 2545613 +[9] +, Homological algebra in bivariant K-theory and other triangulated categories. I, Tri- +angulated categories (Thorsten Holm, Peter Jørgensen, and Rapha¨el Rouqier, eds.), London +Math. Soc. Lecture Note Ser., vol. 375, Cambridge Univ. Press, Cambridge, 2010, pp. 236–289. +MR 2681710 + + +34 +RALF MEYER AND RYSZARD NEST + +[10] Amnon Neeman, Triangulated categories, Annals of Mathematics Studies, vol. 148, Princeton +University Press, Princeton, NJ, 2001. MR 1812507 +[11] Gunnar Restorff, Classification of Cuntz–Krieger algebras up to stable isomorphism, J. Reine +Angew. Math. 598 (2006), 185–210, doi: 10.1515/CRELLE.2006.074. MR 2270572 +[12] +, Classification of Non-Simple C∗-Algebras, Ph.D. Thesis, Københavns Universitet, +2008, http://www.math.ku.dk/~restorff/papers/afhandling_med_ISBN.pdf. +[13] Mikael Rørdam, Classification of extensions of certain C∗-algebras by their six term exact +sequences in K-theory, Math. Ann. 308 (1997), no. 1, 93–117, doi: 10.1007/s002080050067. +MR 1446202 +[14] Steven Vickers, Topology via logic, Cambridge Tracts in Theoretical Computer Science, vol. 5, +Cambridge University Press, Cambridge, 1989. MR 1002193 + +Mathematisches Institut and Courant Research Centre “Higher Order Structures”, +Georg-August Universit¨at G¨ottingen, Bunsenstraße 3–5, 37073 G¨ottingen, Germany +E-mail address: rameyer@uni-math.gwdg.de + +Københavns Universitets Institut for Matematiske Fag, Universitetsparken 5, 2100 +København, Denmark +E-mail address: rnest@math.ku.dk + + diff --git a/papers/project_paper_1_relativity/references/Bombelli2009_source.tar.gz b/papers/project_paper_1_relativity/references/Bombelli2009_source.tar.gz new file mode 100644 index 00000000..19caeded Binary files /dev/null and b/papers/project_paper_1_relativity/references/Bombelli2009_source.tar.gz differ diff --git a/papers/project_paper_1_relativity/references/Loomis2018.pdf b/papers/project_paper_1_relativity/references/Loomis2018.pdf new file mode 100644 index 00000000..7fb1bef8 --- /dev/null +++ b/papers/project_paper_1_relativity/references/Loomis2018.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9d0c3dd63c265a76b970b47a8ee6a08e9302270feb78d39e30f5996f0bf9079f +size 140066 diff --git a/papers/project_paper_1_relativity/references/Loomis2018.txt b/papers/project_paper_1_relativity/references/Loomis2018.txt new file mode 100644 index 00000000..7041c4a7 --- /dev/null +++ b/papers/project_paper_1_relativity/references/Loomis2018.txt @@ -0,0 +1,1035 @@ +arXiv:1709.00064v2 [gr-qc] 25 Sep 2017 + +September 2017 + +Suppression of non-manifold-like sets +in the causal set path integral + +S. P. Loomis∗ and S. Carlip† + +Department of Physics +University of California +Davis, CA 95616 +USA + +Abstract + +While it is possible to build causal sets that approximate spacetime +manifolds, most causal sets are not at all manifold-like. We show that a +Lorentzian path integral with the Einstein-Hilbert action has a phase +in which one large class of non-manifold-like causal sets is strongly +suppressed, and suggest a direction for generalization to other classes. +While we cannot yet show our argument holds for all non-manifold-like +sets, our results make it plausible that the path integral might lead to +emergent manifold-like behavior with no need for further conditions. + +∗email: sloomis@ucdavis.edu +†email: carlip@physics.ucdavis.edu + + +1. +Introduction + +The causal set program offers a simple, elegant picture of spacetime as a discrete set of points, +characterized solely by their causal relations. For all its elegance, though, causal set theory has +a potentially fatal flaw. +We know how to construct causal sets that approximate spacetime +manifolds, by starting with a manifold and extracting a Poisson “sprinkling” of points. But such +manifold-like sets are highly atypical; almost all causal sets do not look like any manifold at all. +If causal sets are fundamental, and manifold-like behavior is emergent, a dynamical process must +somehow suppress almost all typical causal sets, leaving only the rare manifold-like ones. Finding +such a process—especially one that has not been artificially constructed merely to achieve this +goal—is not easy. +In this paper, we show that the ordinary path integral with the causal set version of the +(Lorentzian) Einstein-Hilbert action has a phase in which one large class of non-manifold-like +causal sets is strongly suppressed. The class for which we can rigorously show this suppression, +the two-level orders, is itself not “typical”—we certainly do not claim to show that all non- +manifold-like sets are suppressed. But the two-level orders form a fairly large class, one much +larger than the class of manifold-like causal sets. As we discuss in the conclusion, there are +also hints that our methods may extend to more general classes. Our results thus make it more +plausible that the ordinary path integral, with no additional assumptions, may be enough to lead +to emergent manifold-like behavior. +A numerical analysis of two-dimensional causal sets has shown a similar transition between +a phase dominated by non-manifold-like causal sets and one dominated by manifold-like sets [1]. +In one way, that result is stronger than ours, since it accounts for all non-manifold-like sets. On +the other hand, our results are analytic, hold in any dimension, and use the Lorentzian path +integral rather than analytically continuing to Riemannian signature. + +2. +Non-manifold-like causal sets + +The procedure for constructing a manifold-like causal set is well understood [2]. One starts +with a finite-volume region of a manifold with a Lorentzian metric, “sprinkles” points randomly +by a Poisson process, determines the causal relations among these points from the causal structure +of the manifold, and then “forgets” the manifold, keeping only the points and their causal +relations. For a dense enough sprinkling of points, the resulting causal set retains the fundamental +properties of the original manifold: the Alexandrov neighborhoods determine the topology, the +causal relations determine the conformal class of the metric, and the density of points determines +the conformal factor [3,4]. +But such manifold-like causal sets are highly atypical. The “typical” causal set is a Kleitman- +Rothschild (KR) order, a three-level causal set with approximately n/4 points in the “bottom” +and “top” layers and n/2 points in the “middle” layer [5]. In fact, as n → ∞, the proportion of +n-element causal sets that are KR orders goes to one. +Many other non-manifold-like causal sets also occur frequently. There is, in fact, a hierarchy +of classes of non-manifold-like causal sets [6–9]. Each class is characterized by a parameter p + +1 + + +that is the proportion of possible relations that are actualized, and is dominated by causal sets +with a particular number of levels. The dependence of the size of the class on p is not smooth, +but is described by an piecewise continuous function with infinitely many “phase transitions” +characterized by either the creation of new layers or changes in the relative sizes of the layers. +The intricacies of these classes are beyond the scope of this paper—see [8] for details—but it is +sufficient to point out that the class of non-manifold-like causal sets is dominated by three-level +orders, primarily the KR orders, followed by two-level orders and then four-level orders. +In this paper we will focus on the simplest case of two-level orders. Though these are not +as dominant as the three-level orders, they still form a significant part of the collection of non- +manifold like causal sets. + +3. +Causal set path integrals + +To define a path integral for causal sets, we need two ingredients: an appropriate generaliza- +tion of the Einstein-Hilbert action and a discrete version of an integration measure. The action +we shall use, the Benincasa-Dowker action, was introduced in [10]. For a causal set C with n +elements, it takes the general form [11,12] + +1 +ℏS(C) = µ + +� + +n + + +kmax +� + +k=0 +λkNk + +� + +(3.1) + +where µ and λk are appropriately chosen parameters and Nk denotes the number of pairs of +elements {x, y} ⊂ C such that the cardinality of the set {z ∈ C : x ≺ z ≺ y} is equal to k. The +upper limit kmax can be finite or infinite, though it has a lower bound of ⌊2 + d + +2⌋, where d is the +target spacetime dimension. +Eq. (3.1) replicates the Einstein-Hilbert action in the following sense. Suppose we construct +a causal set by Poisson sprinkling points into a manifold of the target dimension. Then for a +high enough sprinkling density and the correct choices of µ and λk, S(C) is equal to the Einstein- +Hilbert action on average. The specific definitions of µ and λk are complicated, but for d = 4 +and kmax = 3 we have + +1 +ℏS(C) = +� l + +lp + +�2 +(n − N0 + 9N1 − 16N2 + 8N3) +(3.2) + +where lp is the Planck length and l is a length scale determined by the sprinkling density of +events into the spacetime. +For our “integration measure” we shall simply sum over causal sets. As in causal dynamical +triangulations [13], we should perhaps include a combinatorial weight to avoid overcounting +causal sets with special symmetries, but that will not affect our conclusions. The Lorentzian +partition function over any particular class C of causal sets is then + +Z[µ, λ0] = +� + +C∈C +exp +� i + +ℏS(C) +� += +� + +C∈C +exp + +� + +iµ + +� + +n + + +kmax +� + +k=0 +λkNk + +�� + +(3.3) + +2 + + +We will be interested in the large n behavior of this quantity; for a manifold-like causal set with +a fixed sprinkling density, this is the large volume limit. + +4. +Suppression of two-level orders + +For this paper we focus on two-level orders, that is, causal sets C of size n such that there +are no three distinct elements x, y, z ∈ C satisfying x ≺ y ≺ z. This means that Nk = 0 for +k > 0. Intuitively, such sets have only two “moments of time,” and clearly do not resemble +manifolds. As we have mentioned, while they are less common than the three-level KR orders, +two-level orders are still much more common than manifold-like causal sets, and they threaten +to dominate the path integral. +For any n-element causal set, N0 can be no larger than Nmax = n(n−1) + +2 +. We classify such sets +by the proportion 0 ≤ p ≤ 1 of relations, given by N0 = pNmax. For fixed n, p is a discrete +parameter, but in the limit of large n we can approximate it as continuous. The utility of this +classification is that the Benincasa-Dowker action is constant over the class of two-level sets with +a fixed p. Denoting such a class by Cp,n, we can write the partition function over two-level orders +of size n as + +Z[µ, λ0] = +� +dp |Cp,n|eiS(p)/ℏ = eiµn +� 1 + +0 +dp |Cp,n| exp +�1 + +2iµλ0pn2 + o(n2) +� +(4.1) + +where |Cp,n| is the cardinality of the class Cp,n. Here we have written Nmax = 1 + +2n2 + o(n2), where +o(n2) denotes terms subleading to n2, which will be negligible in the large n limit. +To calculate |Cp,n| we consider a decomposition into classes Cq,p,n where we put qn of the +elements in the “top” level and (1−q)n in the “bottom” level. Let us denote α = q(1−q), where +α ≤ 1 + +4 since 0 ≤ q ≤ 1. From the structure of the system, there can be at most αn2 relations—the +maximum occurs when every “bottom” element is related to every “top” element—so from the +definition of p, we have α ≥ 1 + +2p. This in turn implies that p ≤ 1 + +2. +The number of ways to choose pNmax = 1 + +2pn(n − 1) pairs from the possible αn2 relations is + +|Cq,p,n| = +� +αn2 + +1 +2pn(n − 1) + +� +(4.2) + +With both arguments large, we can expand the binomial as + +ln |Cq,p,n| =αn2 ln(αn2) − 1 + +2pn2 ln +�1 + +2pn2 +� +− +� +α − 1 + +2p +� +n2 ln +�� +α − 1 + +2p +� +n2 +� ++ o(n2) + += +� +α ln α − 1 + +2p ln +�1 + +2p +� +− +� +α − 1 + +2p +� +ln +� +α − 1 + +2p +�� +n2 + o(n2) +(4.3) + +For 1 + +2p ≤ α ≤ 1 + +4, this is is a monotonically increasing function of α. This means that |Cq,p,n| is +maximized for q = 1 + +2. Now, |Cp,n| is bounded by +���C 1 + +2,p,n +��� ≤ |Cp,n| ≤ +� + +q +|Cq,p,n| +(4.4) + +3 + + +In the large n limit, the upper bound is dominated by the maximal value of q, so + +ln |Cp,n| = ln |C 1 + +2 ,p,n| + o(n2) = 1 + +4h(2p)n2 + o(n2) +� +p ≤ 1 + +2 + +� +(4.5) + +where +h(x) = −x ln x − (1 − x) ln(1 − x) +(4.6) + +is the entropy function. (As we saw above, p ≤ 1 + +2 for two-level sets, so |Cp,n| = 0 for p > 1 + +2.) +Using (4.5), we can write the partition function as + +Z[µ, λ0] = eiµn +� 1/2 + +0 +dp exp +�1 + +2iµλ0pn2 + 1 + +4h(2p)n2 + o(n2) +� +(4.7) + +To simplify notation, we define + +− µλ0 + +2 += β, +2p = x +(4.8) + +Note that 0 ≤ x ≤ 1 and that, from (3.2), β > 0. The exponent in (4.7) is then n2 + +4 E(x), with + +E(x) = −2iβx + h(x) +(4.9) + +We will evaluate the integral by the method of steepest descents.∗ Here we sketch the method +and results; details are given in the appendix. We first find the saddle point: + +E′(x) = 0 = −2iβ − ln x + ln(1 − x) +(4.10) + +⇒ x0 = +e−iβ + +2 cos β = 1 + +2(1 − i tan β), +1 − x0 = +eiβ + +2 cos β = 1 + +2(1 + i tan β) + +The second derivative at x = x0 is + +E′′(x0) = − 1 + +x0 +− +1 + +1 − x0 += −4 cos2 β +(4.11) + +so the direction of steepest descent is x − x0 real. At the saddle point, + +h(x0) = − e−iβ + +2 cos β ln +� e−iβ + +2 cos β + +� +− +eiβ + +2 cos β ln +� +eiβ + +2 cos β + +� += β tan β + ln(2 cos β) + +E(x0) = −2iβx0 + h(x0) = −iβ + ln(2 cos β) +(4.12) + +Remembering that the exponent is n2 + +4 E(x), we have a saddle point contribution of + +Z[µ, λ0] ∼ eiµn + +n + +� +π + +2|E′′(x0)| exp +�n2 + +4 E(x0) +� += +�π + +8 +eiµn + +n cos β exp +�n2 + +4 [−iβ + ln(2 cos β)] +� +(4.13) + +If | cos β| < 1 + +2, the real part of the exponent is negative, and the path integral is exponentially +suppressed. + +∗An earlier attempt to determine the integral in a quadratic approximation failed; we thank Lisa Glaser for +pointing out an algebraic error that invalidated our first approach. + +4 + + +• +• + +• +x− + +0 +1 + +C− +1 +C− +2 + +C− +3 + +Figure 1: Deformed contour through the saddle point at x0 with tan β > 0 + +This is not quite the whole story. The method of steepest descent requires a contour defor- +mation, and we must check that the rest of the contour does not spoil the result. For tan β > 0, +the saddle point is in the lower half plane, and the contour is shown in figure 1. +We show +in the appendix that the remaining pieces of the contour, C− +1 and C− +2 , are also exponentially +suppressed. If, on the other hand, tan β < 0, we must deform the contour into the upper half +plane, and the remaining pieces are not suppressed. We thus conclude that the path integral for +two-level orders is exponentially suppressed at large volume provided that + +tan +� +−µλ0 + +2 + +� +> 0 +and +����cos µλ0 + +2 + +���� < 1 + +2 +⇒ tan +� +−µλ0 + +2 + +� +> +√ + +3 +(4.14) + +We can also carry the analysis one step further. The saddle point approximation (4.13) is +not exact, and one might worry about the higher order terms in the exponent. In the appendix, +we give a rigorous bound: exponential suppression is guaranteed provided that + +tan +� +−µλ0 + +2 + +� +> +�27 + +4 e−1/2 − 1 +�1/2 +≈ 1.759 +(4.15) + +This gives a minimum value of |µλ0| ≈ 2.108, or a scale ℓ ≈ 1.452ℓp in (3.2). + +5. +Discussion + +The program we have described can be summarized as follows: + +1. Identify a class of causal sets that can be divided into subclasses characterized by some +parameters pi such that the action is constant over each subclass. + +2. Count how large each subclass is, to leading order in the size n of the set, as a function of +the parameters pi. + +3. Analytically evaluate the partition function as an integral over pi, and study how it depends +on the parameters µ and λi in the action. + +We have carried this out for a particularly simple case, in which the division into easily +countable subclasses was fairly straightforward. +But there are hints that our results can be + +5 + + +generalized. Once we move beyond two-level orders, the action (3.2) will include contributions +from N1, N2, and N3, greatly complicating the counting. But for sets with only a few levels, +these contributions may be strongly suppressed. +Consider, for example, a KR order, which has approximately n/4 points in a “bottom” level, +n/2 in a “middle” level, and n/4 in a “top” level.† Pick a “bottom” point x and a “top” point +y. Typically, x will link to approximately n/4 points in the middle level. Imagine coloring these +points red, and the remaining points blue. For {x, y} to contribute to N1, y must then link to +exactly one red point in the middle level, along with approximately n/4 blue points. It is easy +to see that the probability of such a pattern goes as a polynomial in n times 2−n/2. The same is +true for contributions to N2 (for which y must link to exactly two red points) and N3 (for which +y must link to exactly three red points). Similar arguments should hold whenever the number +of levels is small. +This suppression should reduce the analysis of KR orders, and perhaps similar few-level +sets, to the form we have already considered, in which only N0 is important. This is still a +preliminary argument, of course. The N0 combinatorics will be different for different orders, +and one must check that the “atypical” few-level causal sets—three-level sets with different +distributions of points or relations from the KR orders, for instance—remain subdominant. Here +the combinatoric results of [8] may prove useful, but much more work is needed. +There are +also subtleties involving the difference between labeled and unlabeled causal sets that require +careful attention [9]. Numerical exploration of distributions of causal sets and relations may shed +additional light on these problems. + +Acknowledgments + +We are very grateful to Lisa Glaser for pointing out a crucial error in an early version of this +work. We also thank David Rideout for helpful conversations. This work was supported in part +by U.S. Department of Energy grant DE-FG02-91ER40674. + +Appendix. +Steepest descent details + +In this appendix we describe some of the details involved in the steepest descent calculation +of section 4. + +Contours + +The integral (4.7) is over the interval 0 < x < 1. For the method of steepest descent, we +must first deform the contour to go through the saddle point in the direction of steepest descent. +The saddle point is x0 = 1 + +2(1 − i tan β) and the direction of steepest descent is x − x0 real, so the +contours are those of figure 2, where the lower branch is applicable for tan β > 0 and the upper +for tan β < 0. + +†More precisely [5], a KR order has between n/4 − n1/2 ln n and n/4 + n1/2 ln n points in the bottom and top +levels, and between n/2 − ln n and n/2 + ln n points in the middle level. + +6 + + +• +• + +• +x− + +0 +1 + +C− +1 +C− +2 + +C− +3 + +• +x+ + +C+ +1 +C+ +2 + +C+ +3 + +Figure 2: Deformed contours through the saddle point at x0 + +Let us first exclude the contour in the upper half plane. Consider C+ +1 . We can write + +x = +iw, +0 < w < 1 + +2| tan β| +(A.1) + +from which, with the branch cuts shown in figure 2, + +ln x = πi + +2 + ln w, +ln(1 − x) = ln +√ + +1 + w2 − i tan−1 w +(A.2) + +with the inverse tangent lying between 0 and π + +2. Hence + +h(x) = −x ln x − (1 − x) ln(1 − x) + += −iw +�πi + +2 + ln w +� +− (1 − iw) +� +ln +√ + +1 + w2 − i tan−1 w +� + += π + +2 w + w tan−1 w − ln +√ + +1 + w2 + imaginary part +(A.3) + +For ℑx > 0, the contribution from the term −2iβx in E(x) is positive, and + +ℜ E = +�π + +2 + tan−1 w + 2β +� +w − ln +√ + +1 + w2 +(A.4) + +For positive real w, this is always positive, so the integral acquires an exponentially large con- +tribution from C+ +1 . This rules out the saddle point approximation for this contour. +Next consider the contour in the lower half plane. On C− +1 , we can write + +x = −iw, +0 ≤ w ≤ 1 + +2 tan β + +ln(−iw) = −πi + +2 + ln w, +ln(1 + iw) = ln +√ + +1 + w2 + i tan−1 w +(A.5) + +7 + + +with the inverse tangent again lying between 0 and π + +2. Then + +h(x) = iw +� +−πi + +2 + ln w +� +− (1 + iw) +� +ln +√ + +1 + w2 + i tan−1 w +� + += π + +2 w + w tan−1 w − ln +√ + +1 + w2 + imaginary part +(A.6) + +and thus +ℜ E = +�π + +2 + tan−1 w − 2β +� +w − ln +√ + +1 + w2 +(A.7) + +For β > π + +2, this is always negative, and the contribution from C− +1 is exponentially suppressed. +For 0 < β < π + +2, the requirement that | cos β| < 1 + +2 limits us to the range π + +3 < β < π + +2. To proceed, +let us determine the maximum value of ℜ E in this range. +Note first that at w = 0, ℜ E = 0 and the derivative + +d(ℜ E) + +dw += π + +2 + tan−1 w − 2β +(A.8) + +is negative, so ℜ E < 0 for small w. The turning point occurs at + +π +2 + tan−1 w − 2β = 0 ⇒ w = − cot 2β = 1 + +2(tan β − cot β) +(A.9) + +For 1 + +2(tan β − cot β) < w < 1 + +2 tan β, ℜ E is increasing, so its maximum in this range will occur +at the endpoint w = 1 + +2 tan β. At that maximum, + +ℜ E = 1 + +2 + +�π + +2 + tan−1 +�1 + +2 tan β +� +− 2β +� +tan β − ln + +� + +1 + 1 + +4 tan2 β +(A.10) + +Treating this quantity as a function of β and using Mathematica [14] to determine its zeros, we +find that it is negative for .9474 < β < π + +2, an interval that includes the full range of interest. +Hence ℜ E(w) < 0 for any β in the range π + +3 < β < π + +2, and the contribution of the contour C− +1 is +again exponentially suppressed. +The contour C− +2 is basically a reflection, and gives the identical suppression. Let + +x = 1 − iv, +0 ≤ v ≤ 1 + +2 tan β +(A.11) + +Then + +h(x) = −(1 − iv) +� +ln +√ + +1 + v2 − i tan−1 v +� +− iv +�πi + +2 + ln v +� + += π + +2 v + v tan−1 v − ln +√ + +1 + v2 + imaginary part +(A.12) + +and +ℜ E = +�π + +2 + tan−1 v − 2β +� +v − ln +√ + +1 + v2 +(A.13) + +8 + + +which exactly matches (A.7). This match is not accidental; it follows from the fact that + +ℜh(1 − iv) = ℜh(iv) = ℜh(−iv) + +as long as we stay on the same branch of the logarithm. + +Error estimates + +The integral (4.13) is based on a quadratic approximation to E(x). In this case, we can also +get control over the errors. Let x = 1 + +2(1 − u). It is then easy to check that for n ≥ 2, + +dnh +dun = −1 + +2 +(n − 2)! +(1 − u)n−1 − (−1)n + +2 +(n − 2)! +(1 + u)n−1 +(A.14) + +Now expand E(x) around x0. Since u0 = i tan β is imaginary, the two terms in (A.14) evaluated +at x0 are complex conjugates; the odd derivatives are imaginary, while the even derivatives are +real. The Taylor expansion for E(x) around x0, with x − x0 real, is then + +ℜE(x) = ℜE(x0) − + +∞ +� + +n=1 + +22n + +2n(2n − 1)[cos2n−1β][cos(2n − 1)β] (x − x0)2n +(A.15) + +where (4.10) has been used to evaluate ℜ(1 − u0)−(2n−1). Hence +��ℜ(E(x) − E(x0) + 2 cos2 β(x − x0)2) +�� + +≤ + +∞ +� + +n=2 + +22n + +2n(2n − 1)| cos2n−1β|| cos(2n − 1)β| (x − x0)2n + +≤ + +∞ +� + +n=2 + +22n + +2n(2n − 1) + +�1 + +2 + +�4n−1 += + +∞ +� + +n=2 + +2−2n + +n(2n − 1) +(A.16) + +using the facts that | cos β| ≤ 1 + +2 and |x − x0| ≤ 1 + +2. The sum evaluates to + +3 +2 ln 3 + +2 + 1 + +2 ln 1 + +2 − 1 + +4 ≈ 0.0116 + +We can thus state, for instance, that on the line 0 < ℜx < 1, ℑx = − i + +2 tan β with tan β > 0—that +is, the line through the saddle point x0—the exponent E(x) is negative as long as + +| cos β| < 2 · 3−3/2e−1/4 ≈ 0.4942 +(A.17) + +which in turn yields (4.15). We do not know whether this is a sharp limit. + +References + +[1] L. Glaser and S. Surya, Class. Quant. Grav 33 (2016) 065003, arXiv:1410.8775. + +9 + + +[2] L. Bombelli, J. Lee, D. Meyer, and R. Sorkin, Phys. Rev. Lett. 59 (1987) 521. + +[3] L. Bombelli and D. A. Meyer, Phys. Lett. A141 (1989) 226. + +[4] S. Major, D. Rideout, and S. Surya, J. Math. Phys. 48 (2007) 032501, arXiv:gr-qc/0604124. + +[5] D. J. Kleitman and B. L. Rothschild, Trans. Amer. Math. Soc. 205 (1975) 205. + +[6] D. Dhar, J. Math. Phys. 19(8) (1978) 1711. + +[7] D J. Kleitman and B. L. Rothschild, Physica 96A (1979) 254. + +[8] H J. Pr¨omel, A. Steger, and A. Taraz, J. Combin. Theory, Series A 94 (2001) 230. + +[9] J. Henson, D. P. Rideout, R. D. Sorkin, and S. Surya, arXiv:1504.05902. + +[10] D. M. T. Benincasa and F. Dowker, Phys. Rev. Lett. 104 (2010) 181301, arXiv:1001.2725. + +[11] F. Dowker and L. Glaser, Class. Quant. Grav 30 (2013) 195016, arXiv:1305.2588. + +[12] L. Glaser, Class. Quant. Grav 31 (2014) 095007, arXiv:1311.1701. + +[13] J. Ambjørn, J. Jurkiewicz, and R. Loll, Nucl. Phys. B610 (2001) 347, arXiv:hep-th/0105267. + +[14] Wolfram Research, Inc., Mathematica, Version 11.1, Champaign, IL (2017). + +10 + + diff --git a/papers/project_paper_1_relativity/references/Loomis2018_source.tar.gz b/papers/project_paper_1_relativity/references/Loomis2018_source.tar.gz new file mode 100644 index 00000000..7bc84c5d Binary files /dev/null and b/papers/project_paper_1_relativity/references/Loomis2018_source.tar.gz differ diff --git a/papers/references/Bombelli2009.pdf b/papers/references/Bombelli2009.pdf new file mode 100644 index 00000000..28be1f86 --- /dev/null +++ b/papers/references/Bombelli2009.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:7314322f3517e9523e98a59a35a263db650a0659edd0df1d174e084606884119 +size 461293 diff --git a/papers/references/Bombelli2009.txt b/papers/references/Bombelli2009.txt new file mode 100644 index 00000000..1d7a654e --- /dev/null +++ b/papers/references/Bombelli2009.txt @@ -0,0 +1,3668 @@ +arXiv:0810.0096v3 [math.OA] 20 Feb 2012 + +C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: +FILTRATED K-THEORY + +RALF MEYER AND RYSZARD NEST + +Abstract. We define the filtrated K-theory of a C∗-algebra over a finite topo- +logical space X and explain how to construct a spectral sequence that computes +the bivariant Kasparov theory over X in terms of filtrated K-theory. +For finite spaces with totally ordered lattice of open subsets, this spectral +sequence becomes an exact sequence as in the Universal Coefficient Theorem, +with the same consequences for classification. +We also exhibit an example where filtrated K-theory is not yet a complete +invariant. We describe two C∗-algebras over a space X with four points that +have isomorphic filtrated K-theory without being KK(X)-equivalent. For this +space X, we enrich filtrated K-theory by another K-theory functor to a com- +plete invariant up to KK(X)-equivalence that satisfies a Universal Coefficient +Theorem. + +1. Introduction + +1.1. The UCT-problem. One of the main problems in the theory of C∗-algebras +is the computation of the equivariant KK-theory of C∗-algebras endowed with some +extra structure. Here we apply the general techniques developed in [6,9] to the case +of C∗-algebras with a non-trivial ideal lattice. The appropriate version of KK-theory +is Kirchberg’s generalisation of Kasparov theory to C∗-algebras over non-Hausdorff +topological spaces (see [5]). Our goal is to compute it in terms of more manageable +K-theoretic information, generalising the usual Universal Coefficient Theorem that +computes Kasparov’s original theory for C∗-algebras in the bootstrap class by an +exact sequence + +(1.1) +Ext +� +K∗+1(A), K∗(B) +� +֌ KK∗(A, B) ։ Hom +� +K∗(A), K∗(B) +� +. + +The generalisation of the bootstrap class to the case of C∗-algebras with non- +trivial ideal lattice was introduced and studied in [8]. Let us first recall some of +the notation from [8]. Let X be a (usually non-Hausdorff) topological space. A +C∗-algebra over X is a C∗-algebra A endowed with a continuous map Prim(A) → X. +Let C∗alg(X) be the category of C∗-algebras over X; the morphisms in C∗alg(X) are +given by X-equivariant (in obvious sense) ∗-homomorphisms. Taking Kirchberg’s +KK-groups as morphisms and the same objects, we get the category KK(X). It +has a structure of a triangulated category (see [8]). For finite X, the bootstrap +class B(X) is defined as the smallest subcategory of KK(X) that is closed under +suspension, isomorphism, exact triangles, and direct sums and contains all objects +with underlying C∗-algebra C. +General methods from homological algebra suggest to study a homology the- +ory H∗ for C∗-algebras over X, taking values in some Abelian category C. Under +some mild assumptions, the machinery developed in [6, 9] yields an Adams type +spectral sequence which abuts to KK(X; , ), with an E2-term expressed in terms +of H∗. + +2000 Mathematics Subject Classification. 19K35, 46L35, 46L80, 46M18, 46M20. +The second author was supported by the German Research Foundation (Deutsche Forschungs- +gemeinschaft (DFG)) through the Institutional Strategy of the University of G¨ottingen. +1 + + +2 +RALF MEYER AND RYSZARD NEST + +For classification purposes, we need, instead of a spectral sequence, a short exact +sequence of the type (1.1): + +(1.2) +ExtC +� +H∗+1(A), H∗(B) +� +֌ KK∗(X; A, B) ։ HomC +� +H∗(A), H∗(B) +� +, + +and a precise description of the range of H∗. +In this case, given two C∗-algebras A and B over X that belong to the bootstrap +class, an isomorphism of H∗(A) to H∗(B) lifts to a KK(X)-equivalence between A +and B. The results of Eberhard Kirchberg then allow to lift this KK(X)-equivalence +to a ∗-isomorphism A ∼= B, provided A and B are tight, purely infinite, stable, +nuclear and separable; here tightness means that the maps Prim(A) → X and +Prim(B) → X are homeomorphisms (see [5]). It is also shown in [8] that, in the +case when X is finite, any object of the bootstrap class is KK(X)-equivalent to a +tight, purely infinite, stable, nuclear, separable C∗-algebra over X. +Hence the existence of an exact sequence of the form (1.2) for all objects of the +bootstrap class leads to a complete classification of the tight, purely infinite, stable, +nuclear, separable C∗-algebras over X in terms of their image under the functor H∗. + +1.2. Main results. It is relatively easy to construct filtrations on KK which pro- +duce spectral sequences which converge to KK-groups on the bootstrap category +and whose E2-term involves only the K-theory of the quotients K∗(A/J) for the +ideals J corresponding to minimal open subsets of X; an example is the filtration +used in [8, Section 4.1]. +However, this spectral sequence is not very useful for +practical purposes, since it does not degenerate at the E2-level. The second dif- +ferential involves, in particular, the K-theory of various subquotients I/J for the +ideals I ⊂ J ⊂ A and the associated six-term exact sequences in K-theory + +(1.3) + +K0 +� +I +� +� K0 +� +J +� +� K0 +� +J/I +� + +� + +K1 +� +J/I +� + +� + +K1 +� +J +� +� +K1 +� +I +� +. +� + +Also higher differentials do not vanish. +To get a short exact sequence instead, we need to consider more sophisticated ho- +mology theories. The homology theory analysed here is “filtrated K-theory,” which +is in some sense the second approximation to this spectral sequence. Roughly speak- +ing, filtrated K-theory comprises the K-theory of various subquotients together with +all canonical maps between these groups. We will make this definition precise later. +The part of it which involves the exact sequences (1.3) appeared previously in the +work of Gunnar Restorff [11] for Cuntz–Krieger algebras and of Mikael Rørdam [13] +and Alexander Bonkat [2] for extensions of C∗-algebras. The UCT theorem in the +case when the ideal structure is given by I1⊳I2⊳A was obtained by Gunnar Restorff +in his phd-thesis [12], where he introduced an invariant which is a particular case +of filtrated K-theory. +In this paper we prove the following + +Theorem 1.1. The filtrated K-theory satisfies the Universal Coefficient Theorem +and is a complete invariant for C∗-algebras over those finite topological spaces with +a totally ordered lattice of open subsets. + +Note that a C∗-algebra over a space of the type described in this result is essen- +tially the same as a C∗-algebra A together with a finite increasing chain of ideals + +{0} = I0 ⊳ I1 ⊳ I2 ⊳ I3 ⊳ · · · ⊳ In−1 ⊳ In = A. + +We will also show that the spectral sequence associated to the filtrated K-theory +does not collapse in general. Let (X, <) be the partially ordered set, where X = + + +C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY +3 + +{1, 2, 3, 4} with the partial order given by 1, 2, 3 < 4 and no further strict inequalities +between 1, 2, 3. A C∗-algebra over this space is a C∗-algebra A together with an +ideal I and a decomposition of A/I into a direct sum of three orthogonal ideals. + +Theorem 1.2. The filtrated K-theory over (X, <) does not satisfy the Universal +Coefficient Theorem and is not a complete invariant. + +In fact, we give an explicit example of two C∗-algebras A and B over X in the +bootstrap class that have isomorphic filtrated K-theory but are not KK(X)-equivalent. +However, for the particular four-point space X, we still get a complete invari- +ant and a Universal Coefficient Theorem as in (1.2), by adding another K-theory +functor to filtrated K-theory. +It is not clear how to construct such an enriched and still computable filtrated +K-theory for general finite spaces. + +1.3. The general machinery. Now we explain the general machinery behind our +approach. Let us fix a finite topological space X. The first step is the correct +definition of filtrated K-theory. The filtrated K-theory of a C∗-algebra A over X +comprises the Z/2-graded Abelian groups K∗ +� +A(Y ) +� +for all locally closed subsets +Y ⊆ X together with all natural transformations between these groups. The main +issue here is to find all natural transformations. These natural transformations +enter in the definition of the target category of the filtrated K-theory functor and +thus influence the Hom and Ext terms that we expect in the Universal Coefficient +Theorem. +We can guess some of these natural transformations. If U is a relatively open +subset of Y , then A(U) is an ideal in A(Y ), with quotient A(Y )/A(U) = A(Y \ U). +This C∗-algebra extension leads to a natural six-term exact sequence + +(1.4) + +K0 +� +A(U) +� +� K0 +� +A(Y ) +� +� K0 +� +A(Y \ U) +� + +� + +K1 +� +A(Y \ U) +� + +� + +K1 +� +A(Y ) +� +� +K1 +� +A(U) +� +. +� + +These exact sequences provide three types of natural transformations associated to +inclusions of open subsets, restriction to closed subset, and boundary maps. +An obvious source for relations between these natural transformations are morph- +isms of C∗-algebra extensions: since the six-term exact sequences in (1.4) are nat- +ural, each natural morphism of extensions provides some commuting diagrams, +which become relations between our generators. +But do these obvious generators and relations already describe all natural trans- +formations? This turns out to be the case for the spaces studied in this article—both +the positive and the negative examples. Although the authors know no counter- +examples, we do not expect this to be so in general. +The starting point for our study of filtrated K-theory is that the covariant func- +tors A �→ K∗ +� +A(Y ) +� +are representable, that is, they are of the form KK∗(X; RY , A) +for suitable C∗-algebras RY over X—these are the representing objects. Our con- +struction of RY yields commutative C∗-algebras, consisting of C0-functions on suit- +able locally closed subspaces of the order complex of the partial order on X. The +Yoneda Lemma tells us that natural transformations from K∗ +� +A(Y ) +� +to K∗ +� +A(Z) +� + +correspond to KK∗(X; RZ, RY ) ∼= K∗ +� +RY (Z) +� +. These groups are easy enough to +compute in the examples we consider, and turn out to be definable by the concrete +generators and relations mentioned above. +The natural transformations acting on filtrated K-theory form a Z/2-graded pre- +additive category NT . A (countable) module over NT is, by definition, an additive + + +4 +RALF MEYER AND RYSZARD NEST + +functor from NT to the category of (countable) Z/2-graded Abelian groups. By +construction, the filtrated K-theory of any C∗-algebra over X is such a countable +module. Let C be the category of countable NT -modules. This is an Abelian cat- +egory, and filtrated K-theory is a stable homological functor FK from the Kasparov +category KK(X) of C∗-algebras over X to C. +It is easy to check that the functor FK: KK(X) → C is universal in the notation +of [9]. General results on homological ideals in triangulated categories now pro- +duce a cohomological spectral sequence that converges towards KK∗(X; A, B) if A +belongs to the bootstrap class; its E2-term involves Extp +C +� +FK(A), FK(B) +� +. +The main issue is whether the Ext-groups Extp +C +� +FK(A), FK(B) +� +with p ≥ 2 +vanish, so that our spectral sequence degenerates to an exact sequence of the desired +form. +This amounts to checking whether FK(A) has a projective resolution of +length 1 in C. +Already for the non-Hausdorff two-point space considered in [2, 13], the cat- +egory C has infinite cohomological dimension, that is, there are objects that admit +no projective resolution of finite length. But these objects do not belong to the +range of the functor FK. If an NT -module A belongs to the range of FK, then +there are exact sequences + +(1.5) +· · · → A(U) → A(Y ) → A(Y \ U) → A(U) → · · · + +for any Y ∈ LC(X), U ∈ LC(Y ) because of (1.4). But there are NT -modules +without finite length projective resolutions. For totally ordered spaces, an object +of C has a projective resolution of length 1 if and only if it has a projective resolution +of finite length, if and only if the sequences (1.5) are exact, if and only if it is the +filtrated K-theory of some separable C∗-algebra over X, which we can take in the +bootstrap class. +For the four-point counterexample considered in Section 5, we first find a torsion- +free exact module that is not projective, and then use it to find an exact module +without projective resolutions of length 1. Then we find two non-isomorphic objects +of the bootstrap class with the same filtrated K-theory. The idea here is to consider +a certain exact triangle ΣC → A → B → C, which splits on the level of filtrated +K-theory, so that A ⊕ C and B have the same filtrated K-theory. But we can prove +in our concrete example that A ⊕ C and B are not KK(X)-equivalent. +A C∗-algebra over the four-point space X is a C∗-algebra A with a distinguished +ideal I and a direct sum decomposition of A/I as a direct sum of three orthogonal +ideals. +Since both direct sums and extensions of C∗-algebras can be classified +by filtrated K-theory, it is remarkable that the combination of both provides a +counterexample. Incidentally, the space Xop that corresponds to a C∗-algebra A +with a distinguished ideal I and a direct sum decomposition of I as a direct sum of +three orthogonal ideals also leads to a counterexample in a similar fashion. +For the four-point space X above, there is essentially just one module that ought +to be projective but is not. We can add another invariant to filtrated K-theory that +corresponds to this offending module. Since this changes our whole category, it may +lead to further offending modules, which would have to be added in a second step, +and this could, in principle, go on forever. But in the concrete case at hand, we get +projective resolutions of length 1 for all modules over the enriched filtrated K-theory. +As a result, the enriched filtrated K-theory classifies objects of the bootstrap class +over X up to KK(X)-equivalence, and it classifies purely infinite separable nuclear +stable C∗-algebras with primitive ideal space X and simple subquotients in the +bootstrap class. + +1.4. Some basic notation. We shall use the following notation from [8]: +∈∈ we write x ∈∈ C for objects of a category C as opposed to morphisms; + + +C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY +5 + +X topological space, often assumed sober (see [14]); +O(X) set of open subsets of X, partially ordered by ⊆; +LC(X) set of locally closed subsets of X; +LC(X)∗ set of connected, non-empty locally closed subsets of X; +⪯ specialisation preorder on X, defined by x ⪯ y ⇐⇒ {x} ⊆ {y} +A C∗-algebra; +Prim(A) primitive ideal space of A with hull–kernel topology; +I(A) set of closed ∗-ideals in A, partially ordered by ⊆; +C∗alg(X) category of C∗-algebras over X with X-equivariant ∗-homomorphisms +C∗sep(X) full subcategory of separable C∗-algebras over X; +KK(X) Kasparov category of C∗-algebras over X: its objects are separable +C∗-algebras over X, its set of morphisms from A to B is KK0(X; A, B); +B(X) the bootstrap class in KK(X); +iX +Y extension functor C∗alg(Y ) → C∗alg(X) or KK(Y ) → KK(X) for a +subset Y ⊆ X; +ix abbreviation for iX +{x} for x ∈ X; +rY +X restriction functor C∗alg(X) → C∗alg(Y ) or KK(X) → KK(Y ) for a +locally closed subset Y ⊆ X; +Σ suspension ΣA := C0(R, A). + +Roughly speaking, a space is sober if it can be recovered from the lattice O(X). +It is explained in [8, §2.5] why we may restrict attention to such spaces. For finite +spaces, sobriety is equivalent to the separation axiom T0, that is, two points are +equal once they have the same closure. +A C∗-algebra over X is pair (A, ψ) consisting of a C∗-algebra A and a continuous +map ψ: Prim(A) → X. If X is sober, this is equivalent to a map + +ψ∗ : O(X) → I(A), +U �→ A(U), + +that preserves finite infima and arbitrary suprema, that is, + +A +� � + +U∈F +U +� += +� + +U∈F +A(U), +A +� � + +U∈S +U +� += +� + +U∈S +A(U) = + +� + +U∈S +A(U), + +where F ⊆ O(X) is finite and S ⊆ O(X) is arbitrary. In particular, this implies +A(∅) = {0}, A(X) = A, and the monotonicity condition A(U) ⊳ A(V ) for U ⊆ V . +A ∗-homomorphism f : A → B between two C∗-algebras over X is X-equivariant +if f +� +A(U) +� +⊆ B(U) for all U ∈ O(X). +A subset Y ⊆ X is locally closed if and only if Y = U \ V for open subsets +V, U ∈ O(X) with V ⊆ U. Then we define A(Y ) := A(U)/A(V ) for a C∗-algebra A +over X; this does not depend on the choice of U and V by [8, Lemma 2.15]. +If Y ⊆ X is locally closed and A is a C∗-algebra over Y , then we extend A to a +C∗-algebra iX +Y A over X by iX +Y A(Z) := A(Y ∩Z) for Z ∈ LC(X). Conversely, we can +restrict a C∗-algebra B over X to a C∗-algebra rY +X(B) over Y by rY +XB(Z) := B(Z) +for all Z ∈ LC(Y ) ⊆ LC(X). +The category KK(X) is triangulated, with exact triangles coming either from +mapping cone triangles of X-equivariant ∗-homomorphisms or, equivalently, from +semi-split C∗-algebra extensions over X (see [7, 8]). Here an extension is called +semi-split if it splits by an X-equivariant completely positive contraction. +The bootstrap class B(X) is the localising subcategory of KK(X) generated by +the objects ixC for all x ∈ X. That is, it is the smallest class of objects containing +these generators that is closed under suspensions, KK(X)-equivalence, semi-split +extensions, and countable direct sums. + + +6 +RALF MEYER AND RYSZARD NEST + +2. Filtrated K-theory + +Let X be a finite topological space. We do not discuss filtrated K-theory for +C∗-algebras over infinite spaces here. + +Definition 2.1. For a locally closed subset Y ⊆ X, we define a functor + +FKY : KK(X) → AbZ/2, +FKY (A) := K∗ +� +A(Y ) +� +. + +Here Ab denotes the category of Abelian groups and AbZ/2 denotes the category of +Z/2-graded Abelian groups. + +For each Y ∈ LC(X), the functor FKY is stable and homological, that is, it +intertwines the suspension on KK(X) with the translation functor on AbZ/2 (this +functor shifts the grading), and if ΣC → A → B → C is an exact triangle in +KK(X)—this may, for instance, come from a semi-split extension A ֌ B ։ C— +then FKY (A) → FKY (B) → FKY (C) is an exact sequence in AbZ/2. +The functors FKY together form the filtrated K-theory functor. But the latter +also includes its target category, which we now define in a rather abstract way. + +Definition 2.2. For Y, Z ∈ LC(X), let NT ∗(Y, Z) be the Z/2-graded Abelian +group of all natural transformations FKY ⇒ FKZ. The composition of natural +transformations provides a product + +NT i(Y, Z) × NT j(W, Y ) → NT i+j(W, Z), +f, g �→ f ◦ g, + +which is associative and additive in each variable. +We let NT be the Z/2-graded category whose object set is LC and whose morph- +ism space Y → Z is NT ∗(Y, Z). The Abelian group structure on these morphism +spaces turns this into a pre-additive category. + +Definition 2.3. A module over NT is a grading preserving, additive functor +G: NT → AbZ/2. That is, it consists of a family of Z/2-graded Abelian groups +GY = (GY,0, GY,1) for Y ∈ LC(X) and product maps + +NT i(Y, Z) × GY,j → GZ,i+j +for all Y, Z ∈ LC(X), i, j ∈ Z/2; these product maps are associative, additive in +each variable, and the identity transformations in NT (Y, Y ) act identically on GY +for all Y ∈ LC(X). +Let Mod(NT ) be the category of NT -modules. The morphisms in Mod(NT ) +are the natural transformations of functors or, equivalently, families of grading +preserving group homomorphisms GY → G′ +Y that commute with the actions of NT . +Let Mod(NT )c be the full subcategory of countable modules. + +By construction, the natural transformations FKY ⇒ FKZ in NT ∗(Y, Z) induce +maps FKY (A) → FKZ(A) for all A ∈∈ KK(X). This turns +� +FKY (A) +� + +Y ∈LC(X) into +a module over NT . Furthermore, it is well-known that the K-theory of separable +C∗-algebras such as A(Y ) for A ∈∈ KK(X) is countable. + +Definition 2.4. Filtrated K-theory is the functor + +FK = (FKY )Y ∈LC(X) : KK(X) → Mod(NT )c, +A �→ +� +K∗ +� +A(Y ) +�� + +Y ∈LC(X). + +The target category Mod(NT )c is an important part of this definition because +we will compute groups of morphisms and extensions in this category. +Since A(∅) = {0} for all C∗-algebras over X, we have FK∅ = 0, so that ∅ is a +zero object of NT . Therefore, G∅ vanishes for any NT -module. +If Y ∈ LC(X) is not connected, that is, Y = Y1 ⊔ Y2 with two disjoint relat- +ively open subsets Y1, Y2 ∈ O(Y ) ⊆ LC(X), then A(Y ) ∼= A(Y1) ⊕ A(Y2) for any + + +C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY +7 + +C∗-algebra A over X. Hence FKY (A) ∼= FKY1(A) × FKY2(A). The natural trans- +formations that implement this natural isomorphism correspond to a direct sum +diagram Y ∼= Y1 ⊕ Y2 in NT . Therefore, any NT -module has GY ∼= GY1 ⊕ GY2; +here we use the fact that a functor that is additive on morphisms is also additive +on objects, even if the category in question is only pre-additive. +Since X is finite, any locally closed subset is a disjoint union of its connected +components. This corresponds to a direct sum decomposition Y ∼= +� +j∈π0(Y ) Yj +in NT . Therefore, we lose no information when we replace LC(X) by the subset +LC(X)∗ of non-empty, connected, locally closed subsets. + +2.1. The representability theorem. The representability theorem serves two +purposes. We will first use it to describe the category NT . Later, we use it to +construct geometric resolutions in KK(X). + +Theorem 2.5. Let X be a finite topological space. The covariant functors FKY for +Y ∈ LC(X) are representable, that is, there are objects RY ∈∈ KK(X) and natural +isomorphisms +KK∗(X; RY , A) ∼= FKY (A) = K∗ +� +A(Y ) +� + +for all A ∈∈ KK(X), Y ∈ LC(X). + +Before we prove this theorem in §2.2, we first describe the representing ob- +jects RY explicitly, and we use this to describe the groups of natural transformations +NT ∗(Y, Z) as K-theory groups of certain locally compact spaces. +The construction of RY requires some preparation. We equip X with the spe- +cialisation preorder ⪯ as in [8, §2.7]; recall that x ⪯ y if and only if {x} ⊆ {y}. +Since the topological space X is finite, it carries the Alexandrov topology of the +preorder ⪯, that is, a subset Y ⊆ X is open if and only if x ⪰ y ∈ Y implies x ∈ Y . +Similarly, Y ⊆ X is closed if and only if x ⪯ y ∈ Y implies x ∈ Y , and locally +closed if and only if x ⪯ y ⪯ z and x, z ∈ Y implies y ∈ Y . + +Definition 2.6. Let (X, ⪯) be a partially ordered set. Its order complex is the +geometric realisation of the simplicial set Ch(X) whose n-simplices are the chains +x0 ⪯ x1 ⪯ · · · ⪯ xn in X and whose face and degeneracy maps delete or double an +entry of the chain. + +Equivalently, Ch(X) is the classifying space of the thin category that has object +set X and a morphism x → y whenever x ⪯ y. +The order complex is the main ingredient in the construction of the representing +objects RY for Y ∈ LC(X). +The non-degenerate n-simplices in Ch(X) are the strict chains x0 ≺ · · · ≺ xn +in X. We let SX be the set of all strict chains. For each I = (x0 ≺ · · · ≺ xn) ∈ SX, +we let ∆I be a copy of ∆n; more formally, ∆I = {(t, I) | t ∈ ∆n}. We also let +∆◦ +I ⊆ ∆I be the corresponding open simplex ∆n \ ∂∆n. +The space Ch(X) is obtained from the union � +I∈SX ∆I by identifying ∆I with +the corresponding face in ∆J whenever I, J ∈ SX satisfy I ⊆ J. Thus the underly- +ing set of Ch(X) is a disjoint union + +(2.1) +Ch(X) = +� + +I∈SX +∆◦ +I. + +For I ∈ SX, let min I and max I be the (unique) minimal and maximal elements +in SX, respectively. We define two functions + +m, M : Ch(X) → X + +by mapping points in ∆◦ +I to min I and max I, respectively. This well-defines func- +tions on Ch(X) because of (2.1). + + +8 +RALF MEYER AND RYSZARD NEST + +Lemma 2.7. If Y ⊆ X is closed, then m−1(Y ) is open and M −1(Y ) is closed in +Ch(X). If Y ⊆ X is open, then m−1(Y ) is closed and M −1(Y ) is open. If Y ⊆ X +is locally closed, then m−1(Y ) and M −1(Y ) are locally closed. + +Proof. First we show that M −1(Y ) is closed if Y is closed. If I ∈ SX satisfies +max I ∈ Y , then max J ∈ Y for all J ⊆ I because max J ⪯ max I ∈ Y . Hence +∆I ⊆ M −1(Y ) once M −1(Y ) ∩ ∆◦ +I ̸= ∅, so that M −1(Y ) ∩ ∆I is closed for all +I ∈ SX; this implies that M −1(Y ) is closed. +A similar argument shows that m−1(Y ) is closed in Ch(X) if Y is open. Now +the remaining assertions follow easily because the maps m−1 and M −1 commute +with complements, unions, and intersections. +□ + +More explicitly, if Y ⊆ X is open, then m−1(Y ) is the union of the simplices SX +for all chains x0 ≺ x1 ≺ · · · ≺ xn with x0 ∈ Y and hence x0, . . . , xn ∈ Y . Thus + +m−1(Y ) = Ch(Y ) +if Y ⊆ X is open. + +Similarly, + +M −1(Y ) = Ch(Y ) +if Y ⊆ X is closed. + +Here we identify Ch(Y ) with a subcomplex of Ch(X) in the obvious way. +Let Xop be X with the topology for the reversed partial order ≻; that is, the +open subsets of Xop are the closed subsets of X, and vice versa. We may rephrase +Lemma 2.7 as follows: + +Proposition 2.8. The map (m, M): Ch(X) → Xop × X is continuous. + +Let +R := C +� +Ch(X) +� + +be the C∗-algebra of continuous functions on Ch(X). Since + +Prim R = Prim C +� +Ch(X) +� ∼= Ch(X), + +the map (m, M) turns R into a C∗-algebra over Xop × X. We abbreviate + +S(Y, Z) := m−1(Y ) ∩ M −1(Z) ⊆ Ch(X); + +this is a locally closed subset of Ch(X) by Lemma 2.7 + +Definition 2.9. We let RY be the C∗-algebra over X with + +RY (Z) := R(Y op × Z) = C0 +� +S(Y, Z) +� + +for all Y, Z ∈ LC(X); here Y op denotes Y with the subspace topology from Xop. +Equivalently, we let RY be the restriction of R to Y op × X, viewed as a C∗-algebra +over X via the coordinate projection Y op × X → X. + +We will prove the Theorem 2.5 for this choice of RY in §2.2. Taking this for +granted, we use the concrete description of RY to compute the groups of natural +transformations. By the Yoneda Lemma, natural transformations between the func- +tors FKY come from morphisms between the representing objects. More precisely, + +(2.2) +NT ∗(Y, Z) ∼= KK∗(X; RZ, RY ) ∼= FKZ(RY ) = K∗ +� +RY (Z) +� + += K∗ +� +R(Y op × Z) +� += K∗� +m−1(Y ) ∩ M −1(Z) +� += K∗� +S(Y, Z) +� +. + +By the way, the universal property of Kasparov theory says that it makes no +difference for the natural transformations FKY ⇒ FKZ whether we view these two +functors as defined on C∗sep(X) or KK(X). But since RY only represents FKY on +the level of KK(X), we get KK∗(X; RZ, RY ) and not the space of X-equivariant +∗-homomorphisms RZ → RY . + + +C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY +9 + +We describe S(Y, Z) more explicitly using the closure and boundary operations + +Z := {x ∈ X | there is z ∈ Z with x ⪯ z}, +∂Z := Z \ Z, + +�Y := {x ∈ X | there is y ∈ Y with x ⪰ y}, +�∂Y := �Y \ Y. + +Of course, Z is the closure of Z in X and �Y is the closure of Y in Xop. + +Lemma 2.10. If Y, Z ∈ LC(X), then + +S(Y, Z) = Ch(�Y ∩ Z) +� � +Ch(�Y ∩ ∂Z) ∪ Ch(�∂Y ∩ Z) +� +. + +In particular, + +S(Y, Z) = Ch(Y ∩ Z) \ Ch(Y ∩ ∂Z) +if Y is open, + +S(Y, Z) = Ch(�Y ∩ Z) \ Ch(�∂Y ∩ Z) +if Z is closed, + +S(Y, Z) = Ch(Y ∩ Z) +if Y is open and Z is closed. + +Proof. Let x0 ≺ x1 ≺ · · · ≺ xn be a strict chain in X. The interior of the corres- +ponding simplex belongs to S(Y, Z) if and only if x0 ∈ Y and xn ∈ Z. This implies +xj ∈ �Y and xj ∈ Z for all j, so that the simplex belongs to Ch(�Y ∩Z). Furthermore, +we neither have xj ∈ �∂Y ∩ Z for all j nor xj ∈ �Y ∩ ∂Z for all j because x0 ∈ Y +and xn ∈ Z. Thus the simplex belongs neither to Ch(�Y ∩ ∂Z) nor to Ch(�∂Y ∩ Z). +Conversely, if xj ∈ �Y ∩Z for all j and neither xj ∈ �∂Y ∩Z for all j nor xj ∈ �Y ∩∂Z +for all j, then some xj must belong to Y ∩ Z and some xk must belong to �Y ∩ Z. +Since Y ∩ Z is closed in �Y ∩ Z and �Y ∩ Z is open in �Y ∩ Z, this implies x0 ∈ Y and +xn ∈ Z. This shows that the interior of a simplex belongs to S(Y, Z) if and only if +it is contained in Ch(�Y ∩ Z) +� � +Ch(�Y ∩ ∂Z) ∪ Ch(�∂Y ∩ Z) +� +. +□ + +Lemma 2.10 and (2.2) yield + +NT ∗(Y, Z) ∼= K∗� +S(Y, Z) +� ∼= K∗� +Ch(�Y ∩ Z), Ch(�Y ∩ ∂Z) ∪ Ch(�∂Y ∩ Z) +� +. + +This is the K-theory of a finite CW-pair and hence is always finitely generated as +an Abelian group. +If C is any finite simplicial complex, then its barycentric subdivision is of the +form Ch(X), where X is the partially ordered set of non-degenerate simplices in C. +Thus NT ∗(X, X) = K∗(|C|), so that any finitely generated Abelian group arises +as NT ∗(X, X). As a consequence, special properties of the pre-additive category +NT can only be hidden in its composition. +When we identify NT ∗(Y, Z) ∼= KK∗(X; RZ, RY ), then the composition of nat- +ural transformations corresponds to the Kasparov composition product. This gets +somewhat obscured when we follow the isomorphisms + +KK∗(X; RZ, RY ) ∼= K∗ +� +RY (Z) +� += K∗� +S(Y, Z) +� +. + +To describe the composition of natural transformations in terms of K∗� +S(Y, Z) +� +, we +must first lift elements of K∗� +S(Y, Z) +� +back to KK∗(X; RZ, RY ) and then compose +them. The lifting requires a formula for the natural isomorphism + +(2.3) +KK∗(X; RY , A) → K∗ +� +A(Y ) +� + +that occurs in the Representability Theorem. By the Yoneda Lemma, any such +natural transformation is of the form f �→ f∗(ξY ) for a unique + +ξY ∈ K0 +� +RY (Y ) +� += K0� +S(Y, Y ) +� += K0� +Ch(Y ) +� +. + +The natural transformation in (2.3) is generated by the class of the 1-dimensional +trivial vector bundle over the compact space Ch(Y ) or, equivalently, the class of +the unit element in K0 +� +RY (Y ) +� +. + + +10 +RALF MEYER AND RYSZARD NEST + +In the examples we consider later, all natural transformations turn out to be +products of obvious ones, coming from the K-theory six-term exact sequences (1.4). +To check this, we only have to verify that a given element α of KK∗(X; RZ, RY ) lifts +a given element of K∗� +S(Y, Z) +� +. The isomorphism (2.3) maps α to [ξZ]⊗RZ(Z)α(Z) +in K∗ +� +RY (Z) +� += K∗� +S(Y, Z) +� +, where α(Z) in KK∗ +� +RZ(Z), RY (Z) +� +is obtained +from α by restriction to Z. This product is easy to compute. +To get acquainted with this approach to natural transformations, we compute +some important examples. Let Y ∈ LC(X) and U ∈ O(Y ). Since R is a C∗-algebra +over Xop × X, there is an extension + +(2.4) +RY \U ֌ RY ։ RU + +of C∗-algebras over X. It contains C∗-algebra extensions + +RY \U(Z) ֌ RY (Z) ։ RU(Z) + +for all Z ∈ LC(X). Let Z := Y \ U. The extension (2.4) is semi-split in C∗alg(X) +and hence has a class in KK1(X; RU, RZ) and produces an exact triangle + +(2.5) +ΣRU → RZ → RY → RU + +in KK(X). + +Lemma 2.11. The maps in the extension triangle (2.5) correspond to the natural +transformations FKU[1] ⇐ FKZ ⇐ FKY ⇐ FKU in (1.4). + +Proof. The natural transformation µY +U : FKU ⇒ FKY in (1.4) is induced by the +natural ∗-homomorphism j : A(U) → A(Y ). For A = RU, this map is invertible +because S(U, Y ) = S(U, U) = Ch(U). Hence j(ξU) ∈ K0� +S(U, Y ) +� +is again the +class of the trivial vector bundle on Ch(U); this class corresponds to the natural +transformation µY +U . The restriction map RY ։ RU in (2.4) maps [ξY ] to [ξU]— +recall that both [ξY ] and [ξU] are trivial vector bundles. Hence the restriction map +RY ։ RU and the natural transformation µZ +Y correspond to the same class—the +1-dimensional trivial vector bundle on Ch(U)—in K0� +S(U, Y ) +� +. +Similarly, the natural transformation µZ +Y : FKY ⇒ FKZ is induced by the nat- +ural ∗-homomorphism p: A(Y ) ։ A(Z). +For A = RY , this is the restriction +∗-homomorphism C +� +Ch(Y ) +� +→ C +� +Ch(Z) +� +because S(Y, Y ) = Ch(Y ) and S(Y, Z) = +Ch(Z). Since the restriction of a trivial bundle remains trivial, µZ +Y corresponds +to the trivial 1-dimensional vector bundle on S(Y, Z) = Ch(Z). The embedding +RZ ։ RY restricts to an identity map on Z because S(Z, Z) = S(Z, Y ) = Ch(Z). +Since this maps [ξZ] to the trivial bundle, the embedding RZ ։ RY and µZ +Y both +correspond to the same class—the 1-dimensional trivial vector bundle on Ch(Z)—in +K0� +S(Y, Z) +� +. +Finally, we study the boundary map δU +Z : FKZ ⇒ FKU[1]. +We claim that it +corresponds to the class of the extension RZ ֌ RY ։ RU in KK1(X; RU, RZ). +To prove this, we use that Ch(Y ) is the join of the spaces Ch(U) and Ch(Z), so +that there is a continuous map f : Ch(Y ) → [0, 1] whose fibres over 0 and 1 are +Ch(U) and Ch(Z), respectively. +More precisely, let x0 ≺ x1 ≺ · · · ≺ xn be a strict chain in Y and let ξ be a point +of the corresponding simplex with coordinates (t0, . . . , tn) with t0 + · · · + tn = 1, +that is, ξ = t0x0 + · · · + tnxn. Then there is j ∈ {0, . . . , n} with x0, . . . , xj ∈ U, +xj+1, . . . , xn ∈ Z. We can, therefore, write ξ = tUξU + tZξZ with + +ξU = t0x0 + · · · + tjxj + +tU +∈ Ch(U), +tU = t0 + · · · + tj, + +ξZ = tj+1xj+1 + · · · + tnxn + +tZ +∈ Ch(Z), +tZ = tj+1 + · · · + tn. + + +C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY +11 + +We define a continuous map f : Ch(Y ) → [0, 1] by ξ �→ tZ. We have + +S(U, U) = Ch(U) = f −1(0), +S(Z, Z) = Ch(Z) = f −1(1) + +by construction, and hence + +S(Z, U) = Ch(Y ) \ +� +Ch(U) ⊔ Ch(Z) +� += f −1� +(0, 1) +� +. + +Now we can compute some boundary maps. The boundary map + +K0� +S(Z, Z) +� ∼= K0 +� +RZ(Z) +� +→ K1 +� +RZ(U) +� ∼= K1� +S(Z, U) +� + +maps the class of the trivial bundle [ξZ] to f ∗(δ), where δ denotes a generator +of Z ∼= K1� +(0, 1) +� +; this follows from the naturality of the boundary map. +The +boundary map + +K0� +S(U, U) +� ∼= K0 +� +RU(U) +� +→ K1 +� +RZ(U) +� ∼= K1� +S(Z, U) +� + +for the extension RZ ֌ RY ։ RU maps the class of the trivial bundle [ξU] to +−f ∗(δ), again by naturality of the boundary map. +□ + +Remark 2.12. The proof also describes the classes in K0� +S(U, Y ) +� +, K0� +S(Y, Z) +� +, +and K1� +S(Z, U) +� +that correspond to the natural transformations in (1.4). +The +natural transformations FKU ⇒ FKY and FKY ⇒ FKZ are represented by the +classes of the trivial vector bundles over the compact spaces S(U, Y ) and S(Y, Z); +the natural boundary map FKZ ⇒ FKU[1] is represented by f ∗(δ) for a generator +of K1� +(0, 1) +� +. + +2.2. Proof of Theorem 2.5. We check first that the natural transformation +KK∗(X; RY , A) → K∗ +� +A(Y ) +� +induced by ξY is an isomorphism if Y is the min- +imal open subset Ux containing some point x ∈ X. The adjointness relation + +KK∗(X; ix(A), B) ∼= KK∗ +� +A, B(Ux) +� + +for all B ∈∈ KK(X) established in [8, Proposition 3.12] yields + +KK∗(X; ix(C), B) ∼= KK∗ +� +C, B(Ux) +� += FKUx(B), + +that is, ix(C) represents FKUx. To check that RUx does so as well, we must show +that ix(C) and RUx are KK(X)-equivalent. + +Recall that ix(C) = (C, x), where x denotes the map Prim(C) ∼= {x} +⊆ +−→ X, and + +ix(C)(Z) = + +� +C +if x ∈ Z, +0 +otherwise + +for all Z ∈ LC(X). +Since Ux = {y ∈ X | x ⪯ y}, the preordered set Ux has a minimal point, +namely x. +Therefore, the space Ch(Ux) is starlike and hence contractible in a +canonical way towards x. The path from a point in ∆I for I ∈ SUx to the base +point in ∆x lies in ∆I∪{x}. Since max I ∪ {x} = max I, the contraction preserves +the ideals RUx(V ) for V ∈ O(X), so that we get a homotopy equivalence between +C +� +Ch(Ux) +� +and ix(C) in C∗alg(X). Thus RUx corepresents FKUx as well. It is easy +to see that the natural isomorphism KK∗(X; RUx, ) ∼= FKUx is induced by ξUx. +Let Good ⊆ LC(X) be the set of all Z ∈ LC(X) for which the natural trans- +formation KK∗(X; RZ, A) → FKZ(A) induced by ξZ is an isomorphism. We must +show Good = LC(X). We have just seen that Ux ∈ Good for all x ∈ X. +Let Y ∈ LC(X) and U ∈ O(Y ); we claim that all three of U, Y , and Y \ U +are good once two of them are. This follows from the Five Lemma because the + + +12 +RALF MEYER AND RYSZARD NEST + +maps induced by ξZ for Z = U, Y, Y \ U intertwine the maps in the six-term exact +sequences (1.4) and + +KK0(X; RU, A) +� KK0(X; RY , A) +� KK0(X; RY \U, A) + +� + +KK1(X; RY \U, A) + +� + +KK1(X; RY , A) +� +KK1(X; RU, A) +� + +for any A ∈∈ KK(X); the latter six-term exact sequence is induced by the semi- +split extension (2.5). The commutativity of the relevant diagrams follows from the +computations in the proof of Lemma 2.11 (which do not depend on Theorem 2.5). +The two-out-of-three property of Good implies: + +U, V ∈ O(X), +U, V, U ∩ V ∈ Good +=⇒ +U ∪ V ∈ Good + +because (U ∪ V ) \ U = V \ (U ∩ V ). By induction on the length of U, this implies +that all open subsets of X belong to Good. Since any locally closed subset is a +difference of two open subsets, we conclude that Good = LC(X). This finishes the +proof of Theorem 2.5. + +3. An example + +In this section, we restrict our attention to a special class of spaces, namely, the +spaces X = {1, . . ., n} totally ordered by ≤ for n ∈ N. We let + +[a, b] := {x ∈ X | a ≤ x ≤ b}. + +for a, b ∈ Z. We equip X with the Alexandrov topology, so that the open subsets +are [a, n] for all a ∈ X; the closed subsets are [1, b] with b ∈ X, and the locally +closed subsets are those of the form [a, b] with a, b ∈ X and a ≤ b. Any locally +closed subset of X is connected. + +3.1. Computations with the order complex. Since any subset of X is totally +ordered, the space Ch([a, b]) is just a closed simplex of dimension b − a for any +b ≥ a. We denote the corresponding face of Ch(X) by ∆[a,b]. This is understood +to be empty for a > b. +From now on, we let + +Y = [a1, b1], +Z = [a2, b2], +with 1 ≤ a1 ≤ b1 ≤ n and 1 ≤ a2 ≤ b2 ≤ n. + +Then �Y = [a1, n], �∂Y = [b1 + 1, n], Z = [1, b2], and ∂Z = [1, a2 − 1]. Lemma 2.10 +yields + +S(Y, Z) = ∆[a1,b2] \ +� +∆[a1,a2−1] ∪ ∆[b1+1,b2] +� +. + +Now we distinguish three cases: + +Case 1: If a2 ≤ a1 ≤ b2 ≤ b1, then S(Y, Z) = ∆[a1,b2] is a non-empty closed simplex. +Hence NT ∗(Y, Z) ∼= K∗� +S(Y, Z) +� ∼= Z[0] (this means Z in degree 0). +Case 2: If a2 − 1 ≤ b1, a1 < a2, and b1 < b2, then S(Y, Z) is obtained from a closed +simplex by removing two disjoint, non-empty closed faces. Excision yields +NT ∗(Y, Z) ∼= K∗� +S(Y, Z) +� ∼= Z[1] (this means Z in degree 1). +Case 3: In all other cases, S(Y, Z) is either empty, a difference of two closed sim- +plices, or a difference σ \ (τ1 ∪ τ2) for two non-empty closed faces τ1 and τ2 +of a simplex σ that intersect. Then τ1 ∪ τ2 and σ are both contractible, so +that NT ∗(Y, Z) ∼= K∗� +S(Y, Z) +� ∼= 0. + + +C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY +13 + +Summing up, we get + +(3.1) +NT ∗(Y, Z) = + + + + + + + + +Z[0] +if a2 ≤ a1 ≤ b2 ≤ b1, +Z[1] +if a2 − 1 ≤ b1, a1 < a2, and b1 < b2, +0 +otherwise. + +3.2. Products of natural transformations. Our next task is to identify the +natural transformations that correspond to the generators of the groups in (3.1); +this also allows us to compute products in NT . +First we study the grading preserving transformations that appear in the first +case. We introduce a partial order ≥ and a strict partial order ≫ on LC(X) by + +[a1, b1] ≥ [a2, b2] +⇐⇒ +a1 ≥ a2 and b1 ≥ b2, + +[a1, b1] ≫ [a2, b2] +⇐⇒ +a1 > b2. + +Our computation shows that NT 0(Y, Z) ̸= {0} if and only if Y ≥ Z but not +Y ≫ Z. This is equivalent to Y ∩ Z being non-empty, closed in Y , and open in Z. +Under these assumptions, there is a natural non-zero ∗-homomorphism given by +the composition +µZ +Y : A(Y ) ։ A(Y ∩ Z) ֌ A(Z) +because A(Y ∩ Z) is a quotient of A(Y ) and an ideal in A(Z). The natural trans- +formation FKY ⇒ FKZ induced by µZ +Y maps ξY ∈ FKY,0(RY ), which is the class +of the trivial line bundle over S(Y, Y ) = ∆[a1,b1], to the trivial line bundle over +S(Y, Z) = ∆[a1,b2]. Since this is the generator of FKZ,0(RY ) = K0� +S(Y, Z) +� ∼= Z[0], +the natural transformation µZ +Y generates NT 0(Y, Z). +If Y ≫ Z, then we let µZ +Y : A(Y ) → A(Z) be the zero map, which induces the zero +transformation FKY ⇒ FKZ. With this convention, we get µZ +Y ◦ µY +W = µZ +W for all +Y, Z, W ∈ LC(X) with W ≥ Y ≥ Z, also if W ≫ Z; this equation holds on the level +of ∗-homomorphisms and, therefore, also for the induced natural transformations. +We can sum this up as follows: + +Lemma 3.1. The category NT 0 of grading-preserving natural transformations +FKY ⇒ FKZ for Y, Z ∈ LC(X) is the pre-additive category generated by natural +transformations µZ +Y : FKY ⇒ FKZ for all Y ≥ Z with the relations µZ +Y ◦ µY +W = µZ +W +for W ≥ Y ≥ Z and µZ +Y = 0 for Y ≫ Z. + +This list of generators is longer than necessary. Clearly, we can write any µZ +Y +as a product of the transformations µ[a−1,b] +[a,b] +for 2 ≤ a ≤ b ≤ n and µ[a,b−1] +[a,b] +for +1 ≤ a < b ≤ n. Moreover, these transformations themselves are indecomposable, +that is, they cannot be written themselves as products in a non-trivial way. +Now we turn to the natural transformations of degree 1. For any b ∈ X and any +C∗-algebra A over X, we have a natural C∗-algebra extension + +A([b, n]) ֌ A([1, n]) ։ A([1, b − 1]), + +which generates an odd natural transformation + +δb : FK[1,b−1] ⇒ FK[b,n]. + +Composing with the grading preserving natural transformations µ above, we get a +natural transformation of degree 1 + +(3.2) +δZ +Y : FKY = FK[a1,b1] +µ=⇒ FK[1,a2−1] +δa2 +==⇒ FK[a2,n] +µ=⇒ FK[a2,b2] = FKZ + +whenever b1 ≥ a2 − 1. +Equation (3.1) predicts that this transformation vanishes if a1 ≥ a2 or b1 ≥ b2. +This can be verified as follows. Vanishing for a1 ≥ a2 is clear because then [a1, b1] ≫ +[1, a2−1]. By the naturality of the boundary map, the transformation in (3.2) agrees + + +14 +RALF MEYER AND RYSZARD NEST + +with the composition of µ: FK[a1,b1] ⇒ FK[a1,a2−1] with the boundary map for the +extension + +(3.3) +A([a2, b2]) ֌ A([a1, b2]) ։ A([a1, a2 − 1]). + +If b1 ≥ b2, then µ[a1,a2−1] +[a1,b1] +factors through the quotient map in (3.3). +But the +composite of two maps in a six-term exact sequence vanishes. +Equation (3.2) produces a natural transformation δZ +Y ∈ NT 1(Y, Z) whenever +a1 < a2, b1 < b2, and a2 − 1 ≤ b1, that is, whenever (3.1) predicts NT 1(Y, Z) +to be non-zero. We claim that δZ +Y generates this group. This follows because the +natural transformation δZ +Y maps the class of the trivial line bundle over S(Y, Y ) to +the generator of K1� +S(Y, Z) +� ∼= Z. +Notice that NT 1([a2, n], Z) = {0} for any Z ∈ LC(X). Since the natural trans- +formation (3.2) above factors through FK[a2,n], any product of two odd natural +transformations vanishes. Thus the category NT is a split extension of NT 0 by +the bimodule NT 1. The bimodule structure on NT 1 is very simple: a product +µZ +Y ◦ δY +W or δZ +Y ◦ µY +W is equal to δZ +W whenever all three natural transformations are +defined, and zero otherwise. + +Example 3.2. To make our constructions more concrete, we now consider the ex- +ample n = 2, which corresponds to extensions of C∗-algebras. There are only three +non-empty locally closed subsets: 1 = [1, 1], 12 = [1, 2], and 2 = [2, 2]. The order +complex is an interval; we label its end points 1 and 2. The map (m, M) from +Ch(X) = [1, 2] to Xop × X maps + +1 �→ (1, 1), +2 �→ (2, 2), +]1, 2[ �→ (1, 2). + +Correspondingly, we have + +S(1, 1) = {1}, +S(1, 2) = ]1, 2[, +S(1, 12) = [1, 2[, + +S(2, 1) = ∅, +S(2, 2) = {2}, +S(2, 12) = {2}, + +S(12, 1) = {1}, +S(12, 2) = ]1, 2], +S(12, 12) = [1, 2]. + +Taking K-theory, we get + +NT (1, 1) = Z[0], +NT (1, 2) = Z[1], +NT (1, 12) = 0, + +NT (2, 1) = 0, +NT (2, 2) = Z[0], +NT (2, 12) = Z[0], + +NT (12, 1) = Z[0], +NT (12, 2) = 0, +NT (12, 12) = Z[0]. + +3.3. Ring-theoretic properties of the natural transformations. We now ob- +serve some general ring-theoretic properties of NT for X = {1, . . . , n} with the +total order. We exclude the trivial case n = 1. We may replace NT by a Z/2- +graded ring by taking the direct sum of NT ∗(Y, Z) for all Y, Z ∈ LC(X)∗ and +defining the product as usual for a category ring. Then NT -modules become Z/2- +graded modules over this Z/2-graded ring, and ring-theoretic notions such as the +Jacobson radical and the balanced tensor product ⊗NT make sense. + +Definition 3.3. Let NT nil ⊆ NT be the subgroup spanned by the natural trans- +formations µZ +Y with Y ̸= Z and δZ +Y with arbitrary Y, Z. +Let NT ss ⊆ NT be the subgroup spanned by the natural transformations µY +Y +with Y ∈ LC(X)∗. + +Lemma 3.4. The subgroup NT nil is the maximal nilpotent ideal in NT , it is the +nilradical and the Jacobson radical of NT . The subgroup NT ss is a semi-simple +subring, and NT decomposes as a semi-direct product NT nil ⋊ NT ss. + +Proof. Since all µY +Y are idempotent, NT ss is a subring isomorphic to ZLC(X)∗ with +pointwise multiplication. It is easy to see that NT nil is an ideal in NT . It is + + +C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY +15 + +nilpotent, that is, NT k +nil = {0} for some k ∈ N, because LC(X)∗ is finite and ≥ +is a partial order on it. Since NT = NT nil ⊕ NT ss as Abelian groups, we get +the desired semi-direct product decomposition. Since the Jacobson radical of NT ss +vanishes, NT nil is both the nilradical and the Jacobson radical of NT . +□ + +We are going to use Lemma 3.4 to characterise the projective NT -modules. This +characterisation involves the following two definitions. + +Definition 3.5. We call an NT -module M exact if the chain complexes + +· · · → M(U) +µY +U +−−→ M(Y ) +µY \U +Y +−−−→ M(Y \ U) +δU +Y \U +−−−→ M(U) → · · · + +are exact for all Y ∈ LC(X), U ∈ O(Y ) as in (1.5). + +Proposition 3.6. Let K ֌ E ։ Q be an extension of NT -modules. If two of the +modules K, E, Q are exact, so is the third one. + +Proof. Given U and Y as above and a module M, let C•(M) be the chain complex + +· · · → M(U)[m] → M(Y )[m] → M(Y \ U)[m] → M(U)[m − 1] → · · · . + +Then C•(K) ֌ C•(E) ։ C•(Q) is an extension of chain complexes. The long +exact homology sequence shows that all three of these chain complexes are exact +once two of them are exact. +□ + +Definition 3.7. Given an NT -module M, we let + +NT nil · M = {x · m | x ∈ NT nil, m ∈ M}, +Mss := M/NT nil · M. + +We call Mss the semi-simple part of M. + +Since the tensor product over NT is right exact, Mss ∼= NT ss ⊗NT M. We need +the following more concrete description of Mss or, equivalently, of NT nil · M. + +Lemma 3.8. Let M be an NT -module and let Y = [a, b] with 1 ≤ a ≤ b ≤ n. +Then + +(NT nil · M)(Y ) = + + + + + + + + + + + + + + + + + + +µY +[a+1,b](M[a + 1, b]) + µY +[a,b+1](M[a, b + 1]) +if a < b < n, + +µY +[a,b+1](M[a, b + 1]) +if a = b < n, + +µY +[a+1,b](M[a + 1, b]) + δY +[1,a−1](M[1, a − 1]) +if 1 < a < b = n, + +µY +[a+1,b](M[a + 1, b]) +if 1 = a < b = n, + +δY +[1,a−1](M[1, a − 1]) +if a = b = n. + +If M is exact, then + +(NT nil · M)(Y ) = + +� +ker +� +δ[a+1,b+1] +[a,b] +: M[a, b] → M[a + 1, b + 1] +� +if b < n, + +ker +� +µ[1,a] +[a,b] : M[a, b] → M[1, a] +� +if b = n. + +Proof. The first assertion holds because any natural transformation FKZ ⇒ FKY +with Z ̸= Y factors through µY +[a+1,b] or µY +[a,b+1] if a < b < n, through µY +[a,b+1] if + +a = b < n, and so on. Here we use that the natural transformations µ[a−1,b] +[a,b] +for + +2 ≤ a ≤ b ≤ n, µ[a,b−1] +[a,b] +for 1 ≤ a < b ≤ n, and δ[a,n] +[1,a−1] for 2 ≤ a ≤ n already +generate NT ∗, that is, all other transformations µZ +Y or δZ +Y with Y ̸= Z can be +written as products of these generators. By the way, these natural transformations +even form a basis for the subquotient NT nil/NT 2 +nil. +Now assume that M is exact. If a = b < n, then + +(NT nil · M)[a, a] = range +� +µ[a,a] +[a,a+1] +� += ker +� +δ[a+1,a+1] +[a,a] +� +. + + +16 +RALF MEYER AND RYSZARD NEST + +Similarly, we get + +(NT nil · M)[n, n] = ker +� +µ[1,n] +[n,n] +� +, +(NT nil · M)[1, n] = ker +� +µ[1,1] +[1,n] +� +. + +Given f1 : A1 → B and f2 : A2 → B and two exact sequences + +A1 +f1 +−→ B +g1 +−→ C1, +A2 +g1f2 +−−−→ C1 +g2 +−→ C2, + +we have + +(3.4) +range(f1) + range(f2) = ker(g1) + range(f2) + += {x ∈ B | g1(x) ∈ range(g1 ◦ f2) = ker(g2)} = ker(g2 ◦ g1). + +If a < b < n, then we apply this to the maps on M induced by f1 = µY +[a+1,b] +and f2 = µY +[a,b+1] with Y = [a, b]. +We get g1 = µ[a,a] +Y +, g1 ◦ f2 = µ[a,a] +[a,b+1], and + +hence g2 = δ[a+1,b+1] +[a,a] +and g2 ◦ g1 = δ[a+1,b+1] +[a,b] +. This yields the desired formula for +(NT nil · M)[a, b] for a < b < n, using the exactness of M. If a < b = n, then we +apply the same reasoning to f1 = µY +[a+1,b] and f2 = δY +[1,a−1]. Here we get g1 = µ[a,a] +Y +as above, g1 ◦ f2 = δ[a,a] +[1,a−1], and hence g2 = µ[1,a] +[a,a] and g2 ◦ g1 = µ[1,a] +[a,b]. This yields +the desired formula for (NT nil · M)[a, b] for a < b = n. +□ + +Remark 3.9. The natural transformation δ[a+1,b+1] +[a,b] +for b < n or µ[1,a] +[a,n] for b = n +is the longest natural transformation out of [a, b] in the following sense: it factors +through δZ +[a,b] or µZ +[a,b] whenever the latter is defined and non-zero. Thus Lemma 3.8 +identifies NT nil·M(Y ) with the largest proper subgroup of M(Y ) that is the kernel +of some δZ +[a,b] or µZ +[a,b]. + +The following proposition is a rather trivial variant of the Nakayama Lemma. +Unlike in the usual Nakayama Lemma, we do not assume the module to be finitely +generated. This is no problem because the relevant ideal NT nil is nilpotent. + +Proposition 3.10. Let M be an NT -module with Mss = 0. Then M = 0. + +Proof. By assumption, M = NT nil · M. By induction, this implies M = NT j +nil · M +for all j ∈ N. Since NT k +nil = 0 for some k, we get M = 0. +□ + +3.4. Characterisation of free and projective modules. + +Definition 3.11. For Y ∈ LC(X), the free NT -module on Y is defined by + +PY (Z) := NT ∗(Y, Z) +for all Z ∈ LC(X). + +An NT -module is called free if it is isomorphic to a direct sum of degree-shifted +free modules PY [j], j ∈ Z/2. + +Theorem 3.12. Let M be an NT -module. Then the following are equivalent: + +(i) M is a free NT -module. +(ii) M is a projective NT -module. +(iii) Mss(Y ) = NT ss ⊗NT M(Y ) is a free Abelian group for all Y ∈ LC(X) and +TorNT +1 +(NT ss, M) = 0. +(iv) M(Y ) is a free Abelian group for all Y ∈ LC(X) and M is exact. + +Here TorNT +1 +denotes the first derived functor of ⊗NT . The first three conditions +remain equivalent when we replace NT by any ring that is a nilpotent extension of +the ring ZN for some N ∈ N. + + +C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY +17 + +Proof. The Yoneda Lemma asserts that Hom(PY , M) ∼= M(Y ) for all Y ∈ LC(X) +and all NT -modules M. Hence free modules are projective, that is, (1)=⇒(2). A +functor of the form M �→ R ⊗S M for a ring homomorphism S → R always maps +free modules to free modules and hence maps projective modules to projective mod- +ules. Furthermore, derived functors like TorNT +1 +automatically vanish on projective +modules. This yields the implication (2)=⇒(3). We are going to prove that (3) +implies (1). +Since Mss(Y ) is a free Abelian group for all Y , Mss is a free module over NT ss ∼= +ZLC(X)∗. Hence P := NT ⊗NT ssMss is a free NT -module. The canonical projection +M → Mss splits by an NT ss-module homomorphism because Mss is free. This +induces an NT -module homomorphism f : P → M because of the adjointness +relation +HomNT (NT ⊗NT ss X, Y ) ∼= HomNT ss(X, Y ). + +We claim that f is invertible, so that M ∼= P is a free module as asserted. We have + +Pss = NT ss ⊗NT NT ⊗NT ss Mss ∼= NT ss ⊗NT ss Mss ∼= Mss. + +Inspection shows that this isomorphism is induced by f. Since the functor M �→ Mss +is right-exact, this implies coker(f)ss = 0 and hence coker(f) = 0 by the Nakayama +Lemma (Proposition 3.10). That is, f is an epimorphism. +Let K := ker(f), then we get an exact sequence of NT -modules K ֌ P ։ M. +The derived functors of NT ss ⊗NT +provide a long exact sequence + +(3.5) +0 → TorNT +1 +(NT ss, M) → Kss → Pss +f−→ +∼ += Mss → 0. + +This exact sequence ends at TorNT +1 +(NT ss, P) = 0 because P is projective. Since +TorNT +1 +(NT ss, M) = 0 by assumption, we conclude that Kss = 0. Hence another +application of the Nakayama Lemma shows that ker(f) = 0 as well. Thus f is +invertible. This finishes the proof of the implication (3)=⇒(1), showing that the +first three conditions are equivalent. Furthermore, our argument so far works for any +split nilpotent extension of ZN for some N ∈ N because this is the only information +about NT that we have used. Nilpotent extensions of the ring ZN always split +because we can lift orthogonal idempotents in nilpotent extensions. +Free NT -modules are exact, and they consist of free Abelian groups by (3.1). +This yields the implication (1)=⇒(4). We are going to prove that (4) implies (3). +This will finish the proof of the theorem. Since we will use this once again later, +we state half of this argument as a separate lemma: + +Lemma 3.13. Let M be an exact NT -module. Then TorNT +1 +(NT ss, M) = 0. + +Proof. Let π: P → M be an epimorphism with a projective NT -module P, and +let K := ker π. Since projective modules are exact and K ֌ P ։ M is a module +extension, Proposition 3.6 shows that K is exact. We still have an exact sequence +as in (3.5). +Since K and P are exact, Lemma 3.8 identifies Kss(Y ) and Pss(Y ) in a natural +way with subspaces of K(Z) and P(Z) for suitable Z; here we use A/ ker(f) ∼= +range(f) for a group homomorphism f : A → B. Since the map K(Z) → P(Z) +is injective, so is the map Kss(Y ) → Pss(Y ). +Hence the map Kss → Pss is a +monomorphism, forcing TorNT +1 +(NT ss, M) = 0 by (3.5). +□ + +To finish the proof of the implication (4)=⇒(3) in Theorem 3.12, it remains to +check that Mss(Y ) is free for all Y if M is exact and M(Y ) is free for all Y . We +use Lemma 3.8 once again to describe Mss(Y ) as the range of a canonical element +in NT ∗(Y, Z) for a suitable Z. Thus Mss(Y ) is isomorphic to a subgroup of M(Z), +which is a free group by assumption. Hence Mss(Y ) is free as well. +□ + + +18 +RALF MEYER AND RYSZARD NEST + +4. Homological algebra in KK(X) + +Let X be a sober topological space. We are going to apply to KK(X) the gen- +eral machinery for doing homological algebra in triangulated categories discussed +in [9]. This theory goes back to the work on relative homological algebra by Samuel +Eilenberg and John Coleman Moore ([4]), which was carried over to the setting of +triangulated categories by Daniel Christensen [3] and Apostolos Beligiannis [1]. + +4.1. An ideal in KK(X). Our starting point is a rough idea of the invariant we +want to use. This rough idea is expressed by a homological ideal in the triangulated +category. The ideal I in KK(X) relevant for us is defined by + +(4.1) +I(A, B) := +� +f ∈ KK(X; A, B) +�� + +f∗ : K∗ +� +A(Y ) +� +→ K∗ +� +B(Y ) +� +vanishes for all Y ∈ LC(X) +� +. + +It makes no difference if we use LC(X) or LC(X)∗ here. +We claim that I is a homological ideal in the triangulated category KK(X); that +is, it is the kernel (on morphisms) of a stable homological functor from KK(X) +to some stable Abelian category; stability means that the functor intertwines the +suspension automorphism on KK(X) with a given suspension automorphism on the +target Abelian category. +Our starting point is a bare form of filtrated K-theory. Recall the functors + +FKY : KK(X) → AbZ/2, +A �→ K∗ +� +A(Y ) +� + +for Y ∈ LC(X) from Definition 2.1 and let + +F := (FKY )Y ∈LC(X) : KK(X) → +� + +Y ∈LC(X)∗ +AbZ/2, +A �→ +� +K∗ +� +A(Y ) +�� + +Y ∈LC(X)∗. + +The target category � +Y ∈LC(X)∗ AbZ/2 of F is Abelian and carries an obvious sus- +pension functor that shifts the Z/2-grading. The functor F is a stable homological +functor, that is, it intertwines the suspension automorphisms and maps exact tri- +angles to long exact sequences. By definition, + +(4.2) +I = +� + +Y ∈LC(X)∗ +ker FKY = ker F, + +that is, f ∈ I(A, B) if and only if F(f) = 0. Hence I is a homological ideal with +defining functor F. +We also have I = ker FK with FK as in Definition 2.4: the two functors F +and FK only differ through their target categories. For the time being, we pretend +that we do not yet know anything about filtrated K-theory beyond the ideal I it +defines. The general machinery will automatically lead us to the functor FK. +As explained in [9], the homological ideal I yields various notions of homological +algebra. The following descriptions of these notions follow from [9, Lemmas 3.2 +and 3.9, Definition 3.21]. +• A morphism f ∈ KK∗(X; A, B) is +– I-epic if the induced maps K∗ +� +A(Y ) +� +→ K∗ +� +B(Y ) +� +are surjective for +all Y ∈ LC(X); +– I-monic if the induced maps K∗ +� +A(Y ) +� +→ K∗ +� +B(Y ) +� +are injective for +all Y ∈ LC(X); +– an I-equivalence if the induced maps K∗ +� +A(Y ) +� +→ K∗ +� +B(Y ) +� +are +bijective for all Y ∈ LC(X). +• A homological functor F : KK(X) → C to some Abelian category C is +I-exact if F(f) = 0 for all f ∈ I; equivalently, F maps I-epimorphisms +to epimorphisms or F maps I-monomorphisms to monomorphisms. + + +C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY +19 + +• An object A ∈∈ KK(X) is +– I-contractible if K∗ +� +A(Y ) +� += 0 for all Y ∈ LC(X); +– I-projective if the functor KK∗(X; A, ) is I-exact; equivalently, I(A, B) = +0 for all B ∈∈ KK(X), or: any I-epimorphism B → A splits (see [9] +for more equivalent characterisations). +• A chain complex + +· · · → An+1 +δn+1 +−−−→ An +δn +−→ An−1 +δn−1 +−−−→ An−2 → · · · + +in KK(X)—that is, An ∈∈ KK(X) and δn ∈ KK(X; An, An−1) for all n ∈ Z, +subject to the condition δn−1 ◦ δn = 0—is I-exact (in some degree n) if the +induced chain complexes of Z/2-graded Abelian groups + +· · · → K∗ +� +An+1(Y ) +� (δn+1)∗ +−−−−−→ K∗ +� +An(Y ) +� (δn)∗ +−−−→ K∗ +� +An−1(Y ) +� +→ · · · + +are exact (in degree n) for all Y ∈ LC(X). +• An I-projective resolution of A ∈∈ KK(X) is an I-exact chain complex + +· · · → P2 +δ2 +−→ P1 +δ1 +−→ P0 +δ0 +−→ A → 0 → · · · + +with I-projective entries Pn for all n ∈ N. +We shall soon see that there are enough I-projective objects in the sense that any +object of KK(X) has an I-projective resolution. Such resolutions are unique up to +chain homotopy equivalence once they exist. +We use projective resolutions to define derived functors (see [9, Definition 3.27]): +just apply the functor to be derived to an I-projective resolution and take homology. +In particular, this yields extension groups Extn +I(A, B) for all A, B ∈∈ KK(X). Un- +like in usual homological algebra, Ext0 +I(A, B) may differ from the morphism space +in KK(X), compare the exact sequence (4.8) in [6]. + +4.2. Enough projective objects. A strategy to find enough projective objects is +outlined in [9, §3.6]. The idea is to study the left adjoint functor FK⊢ +Y of FKY ; this +is defined on P ∈∈ AbZ/2 if there is FK⊢ +Y (P) ∈∈ KK(X) and a natural isomorphism + +(4.3) +Hom +� +P, FKY (B) +� ∼= KK(X; FK⊢ +Y (P), B) + +for all B ∈∈ KK(X). Notice that FK⊢ +Y need not be defined for all P. +Objects of the form FK⊢ +Y (P) are automatically I-projective because the functor +KK(X; FK⊢ +Y (P), ) factors through FKY by (4.3) and vanishes on I by (4.2). +The simplest case to look for FK⊢ +Y (P) is P = Z[0] (this means Z in degree 0). +The defining property of FK⊢ +Y (Z[0]) is a natural isomorphism + +KK(X; FK⊢ +Y (Z[0]), B) ∼= Hom +� +Z[0], FKY (B) +� ∼= FKY,0(B) = K0 +� +B(Y ) +� +. + +In other words, FK⊢ +Y (Z[0]) must represent the covariant functor FKY . Theorem 2.5 +provides such representing objects, and yields the following: + +Proposition 4.1. For any Y ∈ LC(X), the adjoint functor FK⊢ +Y is defined on a +Z/2-graded Abelian group G = G0 ⊕ G1 if G0 and G1 are free and countable. More +precisely, + +FK⊢ +Y + +�� + +i∈I +Z[εi] + +� + += +� + +i∈I +RY [εi], + +where I is a countable set and εi ∈ Z/2 for all i ∈ I. + +Proof. We have just observed that FK⊢ +Y (Z[0]) = RY . Since FKY is stable, this +implies FK⊢ +Y (Z[1]) = RY [1]. It is a general feature of left adjoint functors that they +commute with direct sums. Since countable direct sums exist in KK(X), we get the +existence of FK⊢ +Y on any free countable Z/2-graded Abelian group. +□ + + +20 +RALF MEYER AND RYSZARD NEST + +Corollary 4.2. There are enough I-projective objects in KK(X), and the class of +I-projective objects in KK(X) is generated by the objects RY for Y ∈ LC(X)∗. +More precisely, any I-projective objects is a retract of a direct sum of suspensions +of these objects. + +Proof. This follows from Proposition 4.1 and [9, Proposition 3.37]. +□ + +Often we do not need retracts here, that is, any I-projective object is a direct +sum of suspensions of RY for Y ∈ LC(X)∗; for the totally ordered spaces studied +in §3, this follows from Theorem 3.12. +Since our ideal I is compatible with countable direct sums, the I-contractible +objects form a localising subcategory of KK(X), that is, they form a class NI of +objects that is closed under countable direct sums, retracts, isomorphism, exact +triangles, and suspensions. Furthermore, NI is the complement of the localising +subcategory that is generated by the I-projective objects. These two subcategories +contain much less information than the ideal itself. Roughly speaking, they will be +the same for any reasonable choice of invariant on KK(X) of K-theoretic nature. + +Proposition 4.3. The localising subcategory that is generated by the I-projective +objects is the bootstrap category B(X). It consists of all objects of KK(X) that are +KK(X)-equivalent to a tight, nuclear, purely infinite, stable, separable C∗-algebra +over X whose simple subquotients belong to the bootstrap category B ⊆ KK. + +Proof. By definition, B(X) is the localising subcategory of KK(X) that is generated +by the objects ix(C) for x ∈ X, see [8]. These generators are I-projective because +they represent the functors FKUx, compare the proof of the Representability The- +orem 2.5. The proof of this theorem also shows that the representing objects RY +belong to the triangulated subcategory of KK(X) generated by RUx for x ∈ X and +hence to B(X). Now Corollary 4.2 shows that all I-projective objects belong to +B(X). Hence the localising subcategory they generate is contained in the bootstrap +class. +Conversely, since the generators of the bootstrap class ix(C) are I-projective, +the localising subcategory generated by the I-projective objects must contain the +whole bootstrap class. This yields the first statement. The second one is contained +in [8, Corollary 5.5]. +□ + +4.3. The universality of filtrated K-theory. The next step in the general pro- +gramme is to determine the universal defining functor for I. This functor is char- +acterised by the universal property that it is I-exact and stable homological and +that any I-exact homological functor on KK(X) factors through it uniquely (up to +natural isomorphism). +The advantage of using the universal functor is that it describes I-projective +resolutions and the associated I-derived functors in KK(X) by projective resolutions +and derived functors in its target Abelian category. +This is the crucial step to +compute these derived functors. +In the presence of enough projective objects, [9, Theorem 3.39] characterises the +universal functor by an adjointness property. In our case, this yields: + +Theorem 4.4. The filtrated K-theory functor FK: KK(X) → Mod(NT )c is the +universal I-exact stable homological functor; here Mod(NT )c denotes the category +of all countable graded NT -modules. + +The ring of natural transformations NT comes in automatically at this point. + +Proof. This is best explained as a special case of a general result on certain homo- +logical ideals. Let T be any triangulated category with countable direct sums, and + + +C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY +21 + +let G be an at most countable set of objects of T. Let IG be the stable homological +ideal defined by the functor + +FG : T → +� + +G∈G +AbZ, +A �→ +� +T∗(G, A) +� + +G∈G. + +We assume that FG(A) is countable for all A ∈∈ T. +We are dealing with the case where T = KK(X) and G = {RY | Y ∈ LC(X)∗}; +Theorem 2.5 identifies T∗(RY , A) = KK∗(X; RY , A) ∼= K∗ +� +A(Y ) +� += FKY (A) for +all Y ∈ LC(X)∗, so that IG = I with I as in (4.1). +Viewing G as a full subcategory of T, it becomes a Z-graded pre-additive cat- +egory, so that we get a corresponding category Mod(Gop)c of countable graded right +modules. We can enrich the functor FG to a functor + +F ′ +G : T → Mod(Gop)c + +because the composition in T provides maps + +T∗(G′, A) ⊗ T∗(G, G′) → T∗(G, A) + +for all G, G′ ∈ G, A ∈∈ T, which form a right G-module structure on +� +T∗(G, A) +� + +G∈G. +We claim that the functor F ′ +G is the universal IG-exact functor. +In the case at hand, our description of the natural transformations FKY ⇒ FKZ +in §2.1 means that Mod(Gop)c = Mod(NT )c and F ′ +G = FK is filtrated K-theory +as defined in Definition 2.4. Hence it suffices to establish the claim above to finish +the proof of Theorem 4.4. +To do this, we check the conditions in [9, Theorem 3.39]. Idempotent morphisms +in KK(X) split because this happens in any triangulated category with countable +direct sums (see [10]). +Call F ′ +G(G) = T( , G) for G ∈ G the free Gop-module +on G. Direct sums of free modules are projective, and any object of Mod(Gop)c is a +quotient of a countable direct sum of free modules. Hence Mod(Gop)c has enough +projective objects. Moreover, + +HomGop� +F ′ +G(G), F ′ +G(A) +� ∼= F ′ +G(A)(G) = T(G, A) + +shows that the left adjoint F ⊢ of F := F ′ +G maps F ′ +G(G) to G ∈∈ T. Since the +domain of F ⊢ is closed under suspensions, countable direct sums, and retracts, the +adjoint is defined on all projective modules. Furthermore, F ◦ F ⊢(P) ∼= P holds +for free modules and hence for all projective modules P. Having checked all the +hypotheses of [9, Theorem 3.39], we can conclude that F ′ +G is indeed universal. +□ + +Since FK: KK(X) → Mod(NT )c is universal, [9, Theorem 3.41] now tells us, +roughly speaking, that homological algebra in KK(X) with respect to I is equivalent +to homological algebra in the Abelian category Mod(NT )c: +• An object A of KK(X) is I-projective if and only if FK(A) ∈ Mod(NT )c +is projective and + +KK∗(X; A, B) ∼= HomNT +� +FK(A), FK(B) +� + +for all B ∈∈ KK(X). +Another equivalent condition is that FK(A) ∈ Mod(NT )c is projective +and A belongs to the localising subcategory generated by the I-projective +objects; the latter agrees with the bootstrap class by Proposition 4.3. +• The functor FK and its partially defined left adjoint FK⊢ restrict to an +equivalence of categories between the subcategories of I-projective objects +in KK(X) and of projective objects in Mod(NT )c. +• For any A ∈∈ KK(X), the functors FK and FK⊢ induce bijections between +isomorphism classes of I-projective resolutions of A and isomorphism classes + + +22 +RALF MEYER AND RYSZARD NEST + +of projective resolutions of FK(A) in Mod(NT )c. That is, a projective res- +olution in Mod(NT )c lifts to a unique I-projective resolution in KK(X). +This provides the “geometric resolutions” that are used in connection with +the usual Universal Coefficient Theorem for KK. +• For all n ∈ N, there is a natural isomorphism + +Extn +I(A, B) ∼= Extn +NT +� +FK(A), FK(B) +� +, + +where the right hand side denotes extension groups in the Abelian category +Mod(NT )c. +• For any homological functor G: KK(X) → C, there is a unique right-exact +functor ¯G: Mod(NT )c → C with ¯G ◦ FK(P) = G(P) for all I-projective P. +The left derived functors of G with respect to I are Ln ¯G ◦ FK for n ∈ N, +where Ln ¯G: Mod(NT )c → C denotes the nth left derived functor of ¯G. + +4.4. The Universal Coefficient Theorem. In the general theory, the next step +is to construct a spectral sequence whose E2-term involves the extension groups +Extn +I(A[m], B); it converges—in favourable cases—to KK∗(X; A, B). This spectral +sequence is constructed in [3, 6]. Since we aim for an exact sequence, not for a +spectral sequence, we only need the special case considered in [9, Theorem 4.4]. +This provides the Universal Coefficient Theorem we want under the assumption +that FK(A) has a projective resolution of length 1 in Mod(NT )c: + +Theorem 4.5. Let A, B ∈∈ KK(X). Suppose that FK(A) ∈∈ Mod(NT )c has a +projective resolution of length 1 and that A ∈∈ B(X). Then there are natural short +exact sequences + +Ext1 +NT +� +FK(A)[j + 1], FK(B) +� +֌ KKj(X; A, B) ։ HomNT +� +FK(A)[j], FK(B) +� + +for j ∈ Z/2, where HomNT and Ext1 +NT denote the morphism and extension groups +in the Abelian category Mod(NT )c and [j] and [j + 1] denote degree shifts. + +The bootstrap class appears here because of Proposition 4.3, which identifies it +with the localising subcategory generated by the I-projective objects. + +Corollary 4.6. Let A, B ∈∈ B(X) and suppose that both FK(A) and FK(B) have +projective resolutions of length 1 in Mod(NT )c. Then any morphism FK(A) → +FK(B) in Mod(NT )c lifts to an element in KK0(X; A, B), and an isomorphism +FK(A) ∼= FK(B) lifts to an isomorphism in B(X). + +Proof. The lifting of a homomorphism follows from Theorem 4.5. Given an iso- +morphism f : FK(A) → FK(B), we can lift f and f −1 to elements α and β of +KK0(X; A, B) and KK0(X; B, A), respectively. Since β ◦ α lifts the identity map +on FK(A), the difference id − β ◦ α belongs to Ext1 +NT +� +FK(A)[j + 1], FK(A) +� +. The +latter is a nilpotent ideal in KK(X; A, A) because of the naturality of the exact +sequence in Theorem 4.5. Hence (id − βα)2 = 0, so that β ◦ α is invertible. The +same argument shows that α ◦ β is invertible, so that α is invertible. +□ + +This corollary is what is needed for the classification programme, and it depends +on resolutions having length 1. Conversely, if there is A for which FK(A) has no +projective resolution of length 1, then it is likely that there exist non-isomorphic +B, D ∈∈ B(X) with FK(B) ∼= FK(D). +The following theorem provides such a +counterexample, but under a stronger assumption. + +Theorem 4.7. Let I be a homological ideal in a triangulated category T with enough +I-projective objects. Let F : T → AIT be a universal I-exact stable homological +functor. Suppose that I2 ̸= 0. Then there exist non-isomorphic objects B, D ∈∈ T +for which F(B) ∼= F(D) in AIT. + + +C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY +23 + +Proof. Since I2 ̸= 0, there is A ∈∈ T with I2(A, ) ̸= 0, that is, A is not +I2-projective. The ideal I2 has enough projective objects as well, so that there +is an exact triangle +ΣN2 +γ2 +−→ ˜A2 +α2 +−→ A +ι2 +−→ N2 +with ι2 ∈ I2 and an I2-projective object ˜A2 (this is part of the phantom castle +constructed in [6], where the same notation is used). +Since ι2 ∈ I, this triangle is I-exact and hence provides an extension + +F(N2)[1] ֌ F( ˜A2) ։ F(A) + +in AIT. Even more, this extension splits because ι2 ∈ I2. This follows because the +canonical map +I(A, N2) → Ext1 +I(A, N2[1]) +implicitly used above factors through I/I2 and hence annihilates ι2 (see [6, Equa- +tion (4.9)]). As a result, F( ˜A2) ∼= F(A) ⊕ F(N2)[1]. +But ˜A2 cannot be isomorphic to A ⊕ N2[1]. If this were the case, then A would +be I2-projective, as a retract of the I2-projective object ˜A2. Then I2(A, ) = 0, +contradicting our choice of A. Hence ˜A2 ̸∼= A ⊕ N2[1]. +□ + +If I2 = 0, then the ABC spectral sequence constructed in [6] degenerates at the +third stage, that is, E3 = E∞. But E2 and E3 differ unless projective resolutions +have length 1. Hence the vanishing of I2 is probably not sufficient for isomorphisms +on the invariant to lift because the boundary map d2 on the second stage of the +ABC spectral sequence may provide further obstructions. +Whether or not filtrated K-theory gives rise to projective resolutions of length 1 +depends on the space in question: we will find positive and negative cases below. +Before we turn to examples, we discuss another important issue: does filtrated +K-theory exhaust all of Mod(NT )c? +This is definitely not the case because of +the additional exactness conditions that hold for objects of the form FK(A). The +following result is not optimal but sufficient for our purposes. + +Theorem 4.8. Let G ∈∈ Mod(NT )c have a projective resolution of length 1. Then +there is A ∈∈ B(X) with FK(A) ∼= G, and this object is unique up to isomorphism +in B(X). + +Proof. Any projective resolution of length 1 in Mod(NT )c is isomorphic to one of +the form +· · · → 0 → FK(P1) +FK(f) +−−−−→ FK(P0) → G +for suitable I-projective objects P1, P0 ∈∈ KK(X) and some f ∈ KK0(X; P1, P0). +Here we use that FK restricts to an equivalence of categories between the subcat- +egories of I-projective objects of KK(X) and of projective objects of Mod(NT )c by +the first paragraph of [9, Theorem 3.41]. +We may embed the morphism f in an exact triangle + +ΣA +h−→ P1 +f−→ P0 +g−→ A. + +Since FK(f) is injective, the map f is I-monic; thus g is I-epic and h ∈ I. Therefore, +the long exact sequence for FK applied to the above triangle degenerates to a short +exact sequence +FK(P1) ֌ FK(P0) ։ FK(A). +This yields FK(A) ∼= G as desired. The uniqueness of A is already contained in +Corollary 4.6. +□ + +It remains to understand which objects of the category Mod(NT )c have a pro- +jective resolution of length 1. + + +24 +RALF MEYER AND RYSZARD NEST + +4.5. Resolutions of length 1 in the totally ordered case. We return to the +example of the space X = {1, . . ., n} totally ordered by ≤ studied in §3. Let NT be +the graded pre-additive category of natural transformations described in §3, and let +C = Mod(NT )c be the Abelian category of NT -modules. The following theorem +characterises NT -modules with projective resolutions of length 1: + +Theorem 4.9. Let M ∈∈ C. The following assertions are equivalent: +(i) M = FK∗(A) for some A ∈∈ KK(X); +(ii) M is exact in the sense of Definition 3.5; +(iii) TorNT +i +(NT ss, M) = 0 for i = 1, 2; +(iv) M has a free resolution of length 1 in C; +(v) M has a projective resolution of length 1 in C; +(vi) M has a projective resolution of finite length in C. + +Proof. The exact sequence (1.4) shows that (i) implies (ii). Theorem 4.8 contains +the implication (v)=⇒(i), and the implications (iv)=⇒(v)=⇒(vi) are trivial. We +will show (ii)=⇒(iii)=⇒(iv) and (vi)=⇒(ii), and this will establish the theorem. +First we show that (vi) implies (ii). +Let 0 → Pm → · · · → P0 → M be a +projective resolution of finite length. By a standard “stabilisation” trick, we can +turn this into a free resolution of the same length. Let + +Zj = ker(Pj → Pj−1) ∼= range(Pj+1 → Pj). + +Thus Zm = 0, P0/Z0 ∼= M, and we have exact sequences Zj ֌ Pj ։ Zj−1 +because our chain complex is exact. Since Zm = 0, the exactness of the projective +modules Pm and Proposition 3.6 show recursively that Zj is exact for j = m − +1, m − 2, . . . , 0, so that M is exact. Thus (vi) implies (ii). +Now we prove (ii)=⇒(iii)=⇒(iv). Let P be a countable free module for which +there is an epimorphism π: P ։ M, and let K := ker π. We have an extension +of NT -modules K ֌ P ։ M. Proposition 3.6 shows that K is exact because P +and M are exact. Furthermore, Tori+1(NT ss, M) ∼= Tori(NT ss, K) for all i ≥ 1 +because P is projective. Lemma 3.13 applied to M and K yields Tori(NT ss, M) = 0 +for i = 1, 2 if M is exact, that is, (ii)=⇒(iii). Now assume (iii). The argument above +yields Tor1(NT ss, K) = 0. Since P is projective, the Abelian groups P(Y ) are free +for all Y ∈ LC(X). The exact sequence in (3.5) yields the same for K(Y ). The +criterion in Theorem 3.12.(3) shows that K is projective. +□ + +Now we combine the existence of projective resolutions of length 1 with The- +orem 4.5, which still required this as a hypothesis: + +Theorem 4.10. Let X be the topological space associated to a totally ordered finite +set, and let A and B be C∗-algebras over X. If A ∈∈ B(X), then there is a natural +short exact sequence + +Ext1 +NT +� +FK(A)[1], FK(B) +� +֌ KK∗(X; A, B) ։ HomNT +� +FK(A), FK(B) +� +. + +In particular, any NT -module morphism FK(A) → FK(B) lifts to an element +in KK∗(X; A, B). If both A and B belong to the bootstrap class B(X), then an +isomorphism FK(A) ∼= FK(B) lifts to a KK-equivalence A ≃ B. + +Proof. Use Theorem 4.5 and Corollary 4.6 together with the existence of projective +resolutions of length 1 ensured by Theorem 4.9. +□ + +Theorem 4.11. Let X be the topological space associated to a totally ordered finite +set, and let A and B be tight, purely infinite, stable, nuclear, separable C∗-algebras +over X whose simple subquotients belong to the bootstrap category. Then an iso- +morphism FK(A) ∼= FK(B) lifts to an X-equivariant ∗-isomorphism A ∼= B. + + +C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY +25 + +Furthermore, any countable exact NT -modules is the filtrated K-module of some +tight, purely infinite, stable, nuclear, separable C∗-algebra over X with simple sub- +quotients in the bootstrap category. + +Proof. A nuclear C∗-algebras over X belongs to the bootstrap category B(X) if and +only if its fibres belong to the non-equivariant bootstrap category B (see [8, Corol- +lary 4.13]). For a tight C∗-algebra over X, these fibres are the same as the simple +subquotients. +It is also shown in [8, Corollary 5.5] that any object of B(X) is +KK(X)-equivalent to a tight, nuclear, purely infinite, simple, separable C∗-algebra +over X whose simple subquotients belong to the bootstrap category B. A deep clas- +sification result of Eberhard Kirchberg shows that any KK(X)-equivalence between +such objects lifts to an X-equivariant ∗-homomorphism. Now the first assertion fol- +lows from Theorem 4.10. The second assertion also uses Theorem 4.8. +□ + +5. A counterexample + +Now we let X := {1, 2, 3, 4} with the partial order 1, 2, 3 < 4 and no relation +among 1, 2, 3. Hence the open subsets of X are + +O(X) = +� +∅, {4}, {1, 4}, {2, 4}, {3, 4}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}, {1, 2, 3, 4} +� +, + +that is, a non-empty subset is open if and only if it contains 4. The associated +directed graph is +• 1 + +4 • +� +❑❑ �❑ +❑ + +sss �s +• 2 + +• 3. + +We frequently denote subsets of X simply by 124 := {1, 2, 4}, and so on. +A C∗-algebra over X is a C∗-algebra A with four distinguished ideals + +I1 := A(14), +I2 := A(24), +I3 := A(34), +I4 := A(4), + +such that I1 + I2 + I3 = A and Ii ∩ Ij = I4 for all 1 ≤ i < j ≤ 3 (see [8, Lemma +2.35]). Equivalently, the ideals Ij/I4 for j = 1, 2, 3 decompose A/I4 into a direct +sum of three orthogonal ideals. The other distinguished ideals are + +A(124) = I1 + I2, +A(134) = I1 + I3, +A(234) = I2 + I3. + +Any subset of X is locally closed. But a connected locally closed subset is either +open or one of the singletons {1}, {2}, and {3}. Hence the set of connected locally +closed subsets is + +LC(X)∗ = {4, 14, 24, 34, 124, 134, 234, 1234, 1, 2, 3}. + +The order complex Ch(X) is a graph with four vertices 1, 2, 3, 4 and edges joining +the first three to the last one: + +Ch(X) = + +���� +���� +1 +▲▲▲▲▲ +▲ + +���� +���� +2 +���� +���� +4 + +���� +���� +3 + +rrrrr +r + +Both maps m, M : Ch(X) → X map the vertices to the corresponding points in X. +Whereas M maps the interior of each edge to 4, the map m maps the interior of +the edge [j, 4] to j for j = 1, 2, 3. +Recall that the space of natural transformations FKY ⇒ FKZ is given by + +NT ∗(Y, Z) ∼= K∗� +S(Y, Z) +� +, +S(Y, Z) := m−1(Y ) ∩ M −1(Z) ⊆ Ch(X). + + +26 +RALF MEYER AND RYSZARD NEST + +Y \Z +4 +14 +24 +34 +124 +134 +234 +1234 +1 +2 +3 + +4 +Z +Z +Z +Z +Z +Z +Z +Z +0 +0 +0 + +14 +0 +Z +0 +0 +Z +Z +0 +Z +Z +0 +0 + +24 +0 +0 +Z +0 +Z +0 +Z +Z +0 +Z +0 + +34 +0 +0 +0 +Z +0 +Z +Z +Z +0 +0 +Z + +124 +Z[1] +0 +0 +Z[1] +Z +0 +0 +Z +Z +Z +0 + +134 +Z[1] +0 +Z[1] +0 +0 +Z +0 +Z +Z +0 +Z + +234 +Z[1] +Z[1] +0 +0 +0 +0 +Z +Z +0 +Z +Z + +1234 +Z[1]2 +Z[1] +Z[1] +Z[1] +0 +0 +0 +Z +Z +Z +Z + +1 +Z[1] +0 +Z[1] +Z[1] +0 +0 +Z[1] +0 +Z +0 +0 + +2 +Z[1] +Z[1] +0 +Z[1] +0 +Z[1] +0 +0 +0 +Z +0 + +3 +Z[1] +Z[1] +Z[1] +0 +Z[1] +0 +0 +0 +0 +0 +Z + +Table 1. The ring of natural transformations + +It is straightforward to compute these K-theory groups, and the results are listed +in Table 1. +Here the rows are labelled by Y , the columns by Z. +For instance, +the entry Z at (14, 1) means that NT ∗(14, 1) ∼= Z. +The trivial 1-dimensional +bundle over S(14, 1) generates this group. +Hence Remark 2.12 shows that the +generator is the natural transformation that we get from the quotient map A(14) ։ +A(1). +Similar arguments show that all the natural transformations of degree 0 +are induced by the familiar restriction and extension ∗-homomorphisms for closed +and open subsets. Moreover, the odd natural transformations arise by composing +these ∗-homomorphisms with boundary maps in K-theory long exact sequences. All +relations that they satisfy are predicted by morphisms of extensions and exactness +of the sequences (1.4). +The computations in §3 were based on a description of indecomposable morph- +isms in the category NT ∗. For the space X in question, these are the maps in the +following diagram: + +(5.1) + +14 +i +� + +i +■■■■■■■■ �■ +124 + +i +❑❑❑❑❑❑❑❑ �❑ +1 + +◦❊ +❊❊ +❊ + +δ + +❊❊ �❊ +❊ + +4 + +i +①①①①①①① �① +① +i +� + +i + +❋❋❋❋❋❋❋❋ �❋ +24 + +i +✉✉✉✉✉✉✉ �✉ +✉ + +i +■■■■■■■■ �■ +134 +i +� 1234 + +r +✉✉✉✉✉✉✉✉ �✉ +r +� + +r + +■■■■■■■■■ �■ +2 +◦δ +� 4 + +34 +i +� + +i +✉✉✉✉✉✉✉✉ �✉ +234 + +i +sssssssss �s +3 + +②② ◦② +② + +δ +②② �② +② + +Here we write i for the extension transformation for an open subset, r for the +restriction transformation for a closed subset, and δ for boundary maps in K-theory +long exact sequences. +The indecomposable morphisms in (5.1) provide a minimal set of generators for +the graded ring NT . To describe NT completely, we list the relations. These are +generated by the following: +• the cube with vertices 4, 14, . . . , 1234 is a commuting diagram, that is, all +the commuting squares involving arrows with label i commute; +• the following composite arrows vanish: + +124 +i−→ 1234 +r−→ 3, +134 +i−→ 1234 +r−→ 2, +234 +i−→ 1234 +r−→ 1, + +1 +δ−→ 4 +i−→ 14, +2 +δ−→ 4 +i−→ 24, +3 +δ−→ 4 +i−→ 34; + +• the sum of the three maps 1234 → 4 via 1, 2, and 3 vanishes. + + +C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY +27 + +These relations imply that the diagrams + +124 +r +� + +r +� + +− + +2 + +◦δ +� + +1 +◦δ +� 4 + +134 +r +� + +r +� + +− + +3 + +◦δ +� + +1 +◦δ +� 4 + +234 +r +� + +r +� + +− + +2 + +◦δ +� + +3 +◦δ +� 4 + +anti-commute and that the composite of two odd maps vanishes. It is routine to +check that the universal pre-additive category with these generators and relations +is given by the groups listed in Table 1. +Define NT nil and NT ss as in Definition 3.3: NT nil is the linear span of the +groups NT ∗(Y, Z) with Y ̸= Z and NT ss is spanned by the groups NT ∗(Y, Y ). +Then NT nil is a nilpotent ideal in NT and NT ss ∼= ZLC(X)∗ is a semi-simple +ring. Thus NT nil is the maximal nilpotent ideal in NT and we have a semi-direct +product decomposition NT ∼= NT nil ⋊ NT ss as in Lemma 3.4. +The next task is to describe the submodule M ′ := NT nil · M ⊆ M for an exact +NT -module M. The following computations are done as in the proof of Lemma 3.8, +using (3.4) and that the morphisms in (5.1) generate NT . + +M ′(14) = range +� +i14 +4 : M(4) → M(14) +� += ker +� +r1 +14 : M(14) → M(1) +� +, + +and symmetrically for 24 and 34; + +M ′(124) = range +� +i124 +14 : M(14) → M(124) +� ++ +� +i124 +24 : M(24) → M(124) +� + += ker +� +δ4 +124 : M(124) → M(4) +� +, + +where δ4 +124 denotes a generator of NT 1(124, 4) ∼= Z; symmetry provides M ′(134) +and M ′(234). We have + +M ′(1) = range +� +r1 +1234 : M(1234) → M(1) +� += ker +� +δ234 +1 +: M(1) → M(234) +� +, + +and symmetrically for 2 and 3, and + +M ′(4) = + +3 +� + +j=1 +range +� +δ4 +j : M(j) → M(4) +� += ker +� +i1234 +4 +: M(4) → M(1234) +� +. + +But something goes wrong with M ′(1234). Equation (3.4) yields + +range +� +i1234 +124 : M(124) → M(1234) +� ++ +� +i1234 +134 : M(134) → M(1234) +� + += ker +� +δ14 +1234 : M(1234) → M(14) +� +; + +to take into account the range of i1234 +234 as well, we need an exact sequence containing +δ14 +1234 ◦ i1234 +234 , which is the generator of NT 1(234, 14) ∼= Z. Since there is no such +exact sequence, our method breaks down at this point. +Another symptom but not a cause of problems is that the map δ4 +124 that describes +M ′(124) is not the longest map out of 124: that would be δ34 +124. +As we shall see, the analogues of Theorems 3.12 and 4.9 become false for the +space X. First, there is a non-projective exact module M with free Mss; secondly, +there is a module that has no projective resolution of length 1; thirdly, there are +A, B ∈ B(X) with I2(A, B) ̸= 0. +Hence Theorem 4.7 provides non-isomorphic +objects in the bootstrap class B(X) with isomorphic filtrated K-theory. The con- +struction of these counterexamples follows the above pattern: first we find a counter- +example to Theorem 3.12, which we use to find one for Theorem 4.9, which is then +used to find an example as in Theorem 4.7. +We begin with the unexpected non-projective module. Let PY for Y ∈ LC(X)∗ + +denote the free NT -module on Y , that is, + +PY (Z) = NT ∗(Y, Z), +HomNT (PY , N) ∼= N(Y ) + + +28 +RALF MEYER AND RYSZARD NEST + +for any Y, Z ∈ LC(X)∗ and any NT -module N. A natural transformation FKY ⇒ +FKZ corresponds to an element in NT ∗(Y, Z) ∼= PY (Z) ∼= HomNT (Pz, PY ) and +thus induces a module homomorphism PZ → PY in the opposite direction. Hence +the three arrows 124, 134, 234 → 1234 in (5.1) induce a module homomorphism + +j : P1234 → P 0 := P124 ⊕ P134 ⊕ P234. + +Table 1 shows that there are no module homomorphisms P 0 → P1234, that is, no +non-zero natural transformations from 1234 to 124, 134, or 234. +The crucial observation is that j is a monomorphism, so that P1234 becomes a +submodule of P 0. Since the longest natural transformations out of 1234 are those +to 14, 24 and 34, this follows from the elementary observations that the maps + +NT ∗(1234, j4) → NT ∗(1234 \ j, j4) + +for j = 1, 2, 3 are, respectively, the identity map on Z. +This follows from the +exactness of free modules because NT ∗(j, j4) = 0 by Table 1. +We describe the quotient + +M := P 0/j(P1234) + +by its values M(Y ) for Y ∈ LC(X)∗ as in (5.1): + +(5.2) + +0 +i +� + +i + +❉❉❉❉❉❉❉❉ �❉ +Z + +i + +❋❋❋❋❋❋❋❋ �❋ +Z + +◦● +●●● +●●● +● + +δ +●●●● +●●● +● + +Z[1] + +i +✇✇✇✇✇✇✇ �✇ +✇ +i +� + +i + +●●●●●●●● �● +0 + +i +③③③③③③③ �③ +③ + +i + +❉❉❉❉❉❉❉❉ �❉ +Z +i +� Z2 + +r +①①①①①①① �① +① +r +� + +r + +❋❋ �❋ +❋ +❋ +❋❋ +❋ +❋ +❋ +Z +◦δ +Z[1] + +0 +i +� + +i +③③③③③③③③ �③ +Z + +i +① �① +① +① +① +①① +① +① +① +Z + +✇✇✇◦ +✇ ✇ +✇✇ +✇ + +δ +✇✇✇ +✇ ✇ +✇✇ +✇ + +The boundary maps δ act by isomorphisms on M because M(j4) = 0 for j = +1, 2, 3. The other maps can be understood by writing M(1234) = Z3/⟨(1, 1, 1)⟩ and +M(j) = Z2/⟨(1, 1)⟩ for j = 1, 2, 3 as quotients. The three maps Z → Z2 correspond +to the three coordinate embeddings Z ֌ Z3, the maps Z2 → Z to the projections +Z3 ։ Z2 onto coordinate hyperplanes. +The projective resolution + +(5.3) +0 → P1234 +j−→ P 0 ։ M + +does not split because there exist no non-zero morphisms P 0 → P1234. Hence M +is not projective. But Mss is free, and M is exact because the exact modules form +an exact category and P1234 and P 0 are exact. Thus M is a counterexample to +Theorem 3.12. +The module M is directly related to the problem with describing NT nil·M(1234) +encountered above. Since HomNT (PY , N) ∼= N(Y ) for any NT -module N and any +Y ∈ LC(X)∗, the resolution (5.3) provides an exact sequence + +0 → HomNT (M, N) + +→ N(124) ⊕ N(134) ⊕ N(234) → N(1234) → Ext1 +NT (M, N) → 0, + +so that +Ext1 +NT (M, N) ∼= N(1234)/NT nil · N(1234) ∼= Nss(1234). +Now we use M to construct a counterexample for Theorem 4.9. Let k ∈ N≥2 +and let Mk := M/k · M; that is, we replace Z by Z/k everywhere in (5.2). This +module has a projective resolution of length 2 of the form + +(5.4) +0 → P1234 +(−k,j) +−−−−→ P1234 ⊕ P 0 +(j,k) +−−−→ P 0 ։ Mk, + + +C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY +29 + +where k denotes multiplication by k. Using this resolution, we compute + +Ext2(Mk, P1234) ∼= Z/k, +Ext1(Mk, P1234) ∼= Hom(Mk, P1234) ∼= 0 + +because there are no no-zero morphisms P 0 → P1234. Of course, the generator of +Ext2(Mk, P1234) is the class of the projective resolution (5.4). Hence Mk admits no +projective resolution of length 1 and is a counterexample to Theorem 4.9. +Now we claim that Mk is the filtrated K-theory of some C∗-algebra Ak over X in +the bootstrap class B(X). To begin with, M is the filtrated K-theory of some such +C∗-algebra A by Theorem 4.8. Let Bk be a C∗-algebra in the bootstrap class with +K0(Bk) = Z/k and K1(Bk) = 0; for instance, Bk could be the Cuntz algebra Ok+1. +Then Ak := A ⊗ Bk has filtrated K-theory Mk by the K¨unneth Theorem for the +K-theory of tensor products. + +Theorem 5.1. Let Ak be a C∗-algebra in the bootstrap class with FK(Ak) ∼= Mk +as constructed above. Then Ak is not I2-projective. Hence there exist B, D ∈ B(X) +that are not KK(X)-equivalent but with the same filtrated K-theory. + +Proof. The second assertion follows from the first one using Theorem 4.7 applied +to the bootstrap class B(X) and the restriction of I to B(X). +It remains to prove that Ak cannot be I2-projective. To see this, we lift the +resolution (5.4) to an I-projective resolution + +0 +◦ +�P2 +◦ +�P1 +◦ +�P0 +�Ak + +in B(X) with boundary maps of degree 1, and embed the latter in a phantom tower +(see [6]): + +Ak +N0 +ι1 +0 +� N1 +ι2 +1 +� + +◦✠✠✠ + +�✠✠✠ + +N2 +ι3 +2 +� + +◦✠✠✠ + +�✠✠✠ + +N3 + +◦✠✠✠ + +�✠✠✠ + +N3 + +◦☛☛☛ + +�☛☛☛ + +· · · + +P0 + +π0 + +�✺✺✺✺✺✺ +P1 + +π1 + +�✺✺✺✺✺✺ +� +P2 + +π2 + +�✺✺✺✺✺✺ +� +0 + +�✸✸✸✸✸✸ +� +· · · +� + +The inductive system (Nj, ιj+1 +j +) becomes constant at N3 because Pj = 0 for j ≥ 3. +Since Ak belongs to the bootstrap class, N3 ∼= 0 (see the proof of [6, Proposition +4.5]). This implies N2 ∼= P2. +The composite map ι2 +0 : Ak = N0 → N2 ∼= P2 belongs to I2. Suppose that Ak +were I2-projective. Then ι2 +0 = ι2 +1 ◦ ι1 +0 would vanish, and the long exact homology +sequence would yield that the map ι2 +1 : N1 → N2 must factor through the map +N1 → P0. But + +KK∗(X; P0, P2) ∼= HomNT +� +FK(P0), FK(P2) +� += HomNT (P 0, P1234) = 0. + +Here we have used that filtrated K-theory, by universality, is fully faithful on +I-projective objects and that there are no non-zero module homomorphisms P 0 → +P1234. Since ι2 +1 factors through the zero group, it must be the zero map. But then +the map P1 → N1 must be a split surjection, so that N1 is I-projective. Then the +I-exact triangle ΣAk → ΣN1 → P0 → Ak provides an I-projective resolution of Ak +of length 1, which is impossible because FK(Ak) ∼= Mk has no projective resolution +of length 1. As a consequence, Ak is not I2-projective. +□ + +We can make the two non-equivalent C∗-algebras over X with the same filtrated +K-theory more explicit. One of them is Ak ⊕ ΣR1234, the other one is the mapping +cone of the map ι2 +0 : Ak = N0 → N2 ∼= R1234 in the phantom tower above. Both +have Mk ⊕ P1234[1] as their filtrated K-theory. +This counterexample shows that filtrated K-theory does not yet classify purely +infinite stable nuclear separable C∗-algebras in the bootstrap class. + + +30 +RALF MEYER AND RYSZARD NEST + +Remark 5.2. Refining filtrated K-theory by taking filtrated K-theory with coeffi- +cients does not help. This gets rid of the counterexample Ak constructed above, +but other objects of B(X) without projective resolution of length 1 remain. An ex- +ample is A ⊗ B, where B is a C∗-algebra in the bootstrap class with K∗(B) = Q[0] +such as an appropriate UHF-algebra. Its filtrated K-theory is M ⊗ Q. This also +has cohomological dimension 2, and this is not affected much by taking K-theory +with coefficients because M ⊗ Q is torsion-free. + +5.1. A refined invariant. There are at least two ways to identify the source of +the problem for the space X. The first point of view is that what is missing is an +exact sequence that has the generator α of NT 1(234, 14) as its connecting map. +The map α corresponds to a map ΣR14 → R234 between the representing objects, +which we also denote by α. In the triangulated category KK(X), we can embed the +latter map in an exact triangle + +(5.5) +ΣR14 +α−→ R234 → R12344 → R14. + +The notation R12344 will be explained later. The functors these objects represent +sit in a long exact sequence + +(5.6) +· · · → FK14 → FK12344 → FK234 +α−→ FK14[1] → · · · + +which is precisely what we want. The second point of view is that the troublemaker +is the non-projective module M. Since M has a projective resolution of length 1, +there is a unique object in the bootstrap class with filtrated K-theory M. Actually, +this yields the same object as the first point of view: + +Lemma 5.3. The non-projective module M above agrees with FK(R12344). + +Proof. The map FKY (α) vanishes for almost all Y ∈ LC(X)∗ simply because the +graded groups involved have different parity or one of them vanishes. The only +exception is Y = 14. The group FK14(R14) = NT (14, 14) is generated by the +identity natural transformation. Since α is the generator of NT 1(234, 14), the map +FK14(α) is invertible. +Now we apply FK to the long exact sequence for the given exact triangle. Since +FK(α) vanishes on most Y and is invertible for Y = 14, we can easily compute the +groups FKY (R12344). We get the same groups as for the module M. It remains to +check that the isomorphism can be chosen as an NT -module homomorphism. The +main step is to check that the map + +Z2 ∼= FK124(R12344) ⊕ FK134(R12344) → FK1234(R12344) ∼= Z2 + +is invertible. Together with the known relations between the various natural trans- +formations, this implies the assertion. We omit the details of this computation. +□ + +The representing object R12344 is an algebra of functions on a two-dimensional +simplicial complex, which we do not describe here because it is not illuminating. +The functor that it represents, however, can be described rather nicely as follows. +Let A be a C∗-algebra over X. Pull back the extension A(14) ֌ A(124) ։ A(2) +along the quotient map A(234) ։ A(2) to an extension A(14) ֌ A(12344) ։ +A(234). The object R12344 represents the functor + +(5.7) +KK∗(X; R12344, A) ∼= K∗ +� +A(12344) +� +. + +To see this, two observations are necessary. First, K∗ +� +R12344(12344) +� ∼= Z; the +generator of this group yields a natural transformation between the two functors +in (5.7). Secondly, this natural transformation is invertible. This follows from the +Five Lemma, once we know that it extends the known natural isomorphisms + +KK∗(X; RY , A) ∼= K∗ +� +A(Y ) +� + + +C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY +31 + +for Y = 14 and Y = 234 to a chain map between the long exact sequences that +we get from (5.5) and from the extension A(14) ֌ A(12344) ։ A(234). +This +extension also explains the notation R12344. +Now we augment filtrated K-theory by adding the covariant functor + +B �→ FK12344(B) := K∗ +� +A(12344) +� ∼= KK∗(X; R12344, B). + +The new invariant takes values in the category of countable NT ′-modules, where +NT ′ is the Z/2-graded category whose object set is LC′ := LC(X)∗ ⊔ {12344} and +whose morphisms are the natural transformations between the various filtrated +K-groups, including now also FK12344. These natural transformations can be com- +puted by the Yoneda Lemma: + +NT ′ +∗(Y, Z) ∼= KK∗(X; RZ, RY ) ∼= FKZ(RY ) + +holds for all Y, Z ∈ LC′. The category ring for NT ′ +∗ is simply the ring KK∗(X; R, R) +where +R := +� + +Y ∈LC′ +RY . + +We replace the ideal I in KK(X) studied above by the kernel I′ of the enriched +filtrated K-theory functor + +FK′ : KK(X) → Mod(NT ′)c. + +The same arguments as above show that there are enough I′-projective objects and +that FK′ is the universal I′-exact stable homological functor. +The passage from I to I′ has improved the situation because R12344 has now +been promoted to an I′-projective object and, therefore, ceases to cause trouble. +In principle, something similar can be done in great generality: whenever we have +an object of the Abelian approximation that has a projective resolution of length 1, +we can lift it uniquely to an object of the triangulated category and refine the +ideal by intersecting it with the kernel of the functor this lifted object represents. +However, the policy to quieten troublemakers by promotion has the tendency to +encourage new troublemakers, so that it is not clear whether this general strategy +always resolves all problems after finitely many steps. But in the relatively simple +example at hand, this turns out to be the case. +To check this, we must describe the category NT ′. If Y, Z ∈ LC(X)∗, then +NT ′ +∗(Y, Z) = NT ∗(Y, Z) is given by the table on page 26. Furthermore, if Z ∈ +LC(X)∗, then NT ′ +∗(12344, Z) ∼= FKZ(R12344) = M(Z) by Lemma 5.3, and this is +described in (5.2). The upshot is: +• there are even natural transformations from FK12344 to FK124, FK134, +FK234—the generators of the respective groups of natural transformations— +such that any natural transformation FK12344 ⇒ FKZ with Z ∈ LC(X)∗ + +is a sum of natural transformations that factor through one of these three +and a natural transformation FKij4 ⇒ FKZ; +• the sum of the three natural transformations FK12344 ⇒ FK1234 via FK124, +FK134 and FK234 vanishes, and all other relations follow from these and +the already known ones listed after (5.1). +The exact triangle (5.5) yields a long exact sequence + +· · · → NT ′ +∗+1(Y, 234) +α−→ NT ′ +∗(Y, 14) → NT ′ +∗(Y, 12344) → NT ′ +∗(Y, 234) → · · · , + +which we may use to compute NT ′ +∗(Y, 12344) for all Y ∈ LC′. The map α induces +an isomorphism for Y = 234 and the zero map for all other Y because the source +and target have opposite parity or one of them vanishes. Thus + +Y +4 +14, 24, 34 +124, 134, 234 +1234 +1, 2, 3 +12344 + +NT ′ +∗(Y, 12344) +Z2 +Z +0 +Z[1] +Z[1] +Z + + +32 +RALF MEYER AND RYSZARD NEST + +These groups inherit from M their invariance under permutations of 1, 2, 3. Inspect- +ing composition with natural transformations in NT , we arrive at the following: + +• there are even natural transformations FKj4 ⇒ FK12344 for j = 1, 2, 3, such +that any natural transformation FKY ⇒ FK12344 with Y ∈ LC(X)∗ factors +through one of them; +• the sum of the three natural transformations FK4 ⇒ FK12344 vanishes, +• the natural transformations FKj4 ⇒ FK1234\j via FK12344 vanish; +• all other relations follow from these and the already known ones. + +As one may expect, the basic natural transformations FK14 ⇒ FK12344 ⇒ FK234 +are induced by the maps R234 → R12344 → R14 in the exact triangle (5.5). +The indecomposable morphisms of the new category NT ′ are the maps in the +following diagram: + +14 + +❋❋❋❋❋❋❋ �❋ +124 + +❋❋❋❋❋❋❋ �❋ +1 + +◦❃❃❃❃ + +�❃❃❃❃ + +4 + +⑦ �⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +⑦ +� + +❅❅ �❅ +❅ +❅❅ +❅ +24 +� 12344 +� + +✈✈✈✈✈✈✈ �✈ +✈ + +❍❍❍❍❍❍❍❍ �❍ +134 +� 1234 + +③③③ �③ +③③③③ +� + +❉ �❉ +❉ +❉ +❉ +❉ +❉ +❉ +❉ +2 +◦ +� 4 + +34 + +① �① +①① +①① +①① +① +234 + +①①①①①①① �① +3 + +◦���� + +����� + +The category ring of NT ′ again has the by now familiar structure: it is a split +nilpotent extension of the semisimple algebra NT ′ +ss ∼= ZLC′ spanned by the identity +transformations on the objects and a nilpotent ideal NT ′ +nil that is the subgroup +generated by NT ′(Y, Z) with Y ̸= Z. + +Definition 5.4. A module over NT ′ is exact if it is exact as an NT -module and +the three sequences + +· · · → N∗+1(ij4) → N∗(k4) → N∗(12344) → N∗(ij4) → · · · + +for {i, j, k} = {1, 2, 3} are exact as well. + +The range of the invariant FK′ consists of exact NT ′-modules; the three new +exact sequences are, in fact, equivalent for symmetry reasons, and the extension + +· · · → N∗+1(234) → N∗(14) → N∗(12344) → N∗(234) → · · · + +is built into the definition of FK12344. +Let N be an exact NT ′-module and let N ′ := NT ′ +nil · N. The description of +N ′(14), N ′(1), and N ′(4) is the same as for the category NT , so that these groups +remain kernels of certain maps, as needed. Furthermore, N ′(1234) is the kernel of +the map N(1234) → N(12344)[1] induced by the generator of NT 1(1234, 12344), +so that the problem that appeared for the category NT is cured. +The computation of N ′(124) changes because this group is now the range of the +arrow N(12344) → N(124). But this is part of a long exact sequence because N is +exact, and we get +N ′(124) = ker +� +N(124) → N(34)[1] +� +, + +and similarly for N ′(134) and N ′(234). +Finally, N ′(12344) is the sum of the ranges of the maps N(j4) → N(12344) for +j = 1, 2, 3. Using exactness, we identify this in two steps with the kernel of the +map N(12344) → N(4)[1] induced by the generator of NT ′ +1(12344, 4). +As a result, the submodule NT ′ +nil·N is described using kernels of maps N(Y ) → +N(Z). By the way, these arrows are the longest arrows starting at Y as in Re- +mark 3.9. The same arguments as for totally ordered spaces now show: + + +C∗-ALGEBRAS OVER TOPOLOGICAL SPACES: FILTRATED K-THEORY +33 + +Theorem 5.5. An NT ′-module N is free if and only if it is projective, if and only +if it is exact and N(Y ) is a free group for all Y ∈ LC′. + +Theorem 5.6. An NT ′-module N has a projective resolution of length 1 if and +only if it is exact. + +Theorem 5.7. Let A and B be C∗-algebras over the four-point space X under +consideration. +If A belongs to the bootstrap class B(X), then there is a natural +short exact sequence + +Ext1 +NT ′ +� +FK′(A)[1], FK′(B) +� +֌ KK∗(X; A, B) ։ HomNT ′� +FK′(A), FK′(B) +� +. + +In particular, morphisms FK′(A) → FK′(B) lift to elements in KK∗(X; A, B). If +both A and B belong to the bootstrap class, then an isomorphism FK′(A) ∼= FK′(B) +lifts to a KK(X)-equivalence. + +Corollary 5.8. The map A �→ FK′(A) is a bijection between the set of isomorphism +classes of tight, stable, purely infinite, separable, nuclear C∗-algebras over X with +simple subquotients in the bootstrap class and the set of isomorphism classes of +countable exact NT ′-modules. + +6. Conclusion + +We have obtained a Universal Coefficient Theorem that computes KK∗(X; A, B) +for A in the bootstrap class and X of a very special form, namely, {1, . . . , n} with +the Alexandrov topology from the total order. This Universal Coefficient Theorem +can be used to carry over classification results for simple, nuclear, purely infinite +C∗-algebras to nuclear, purely infinite C∗-algebras with primitive ideal space X, +using filtrated K-theory as the invariant. +For general finite topological spaces X, we still get a spectral sequence that +computes KK∗(X; A, B) using filtrated K-theory, but this spectral sequence need +not degenerate to an exact sequence, so that isomorphisms on filtrated K-theory +need not lift to X-equivariant KK-equivalences. In fact, we have found a counter- +example. At the same time, we were able to fix the counterexample by refining +filtrated K-theory. It is unclear whether such a refinement is available for all finite +topological spaces and how it looks like. + +References + +[1] Apostolos Beligiannis, Relative homological algebra and purity in triangulated categories, J. +Algebra 227 (2000), no. 1, 268–361, doi: 10.1006/jabr.1999.8237. MR 1754234 +[2] Alexander +Bonkat, +Bivariante +K-Theorie +f¨ur +Kategorien +projektiver +Systeme +von +C∗-Algebren, +Ph.D. +Thesis, +Westf. +Wilhelms-Universit¨at +M¨unster, +2002, +http://deposit.ddb.de/cgi-bin/dokserv?idn=967387191 (German). +[3] J. Daniel Christensen, Ideals in triangulated categories: phantoms, ghosts and skeleta, Adv. +Math. 136 (1998), no. 2, 284–339, doi: 10.1006/aima.1998.1735. MR 1626856 +[4] Samuel Eilenberg and John Coleman Moore, Foundations of relative homological algebra, +Mem. Amer. Math. Soc. No. 55 (1965), 39. MR 0178036 +[5] Eberhard Kirchberg, Das nicht-kommutative Michael-Auswahlprinzip und die Klassifikation +nicht-einfacher Algebren, C∗-Algebras (M¨unster, 1999), Springer, Berlin, 2000, pp. 92–141 +(German). MR 1796912 +[6] Ralf Meyer, Homological algebra in bivariant K-theory and other triangulated categories. II, +Tbil. Math. J. 1 (2008), 165–210. MR 2563811 +[7] Ralf Meyer and Ryszard Nest, The Baum–Connes conjecture via localisation of categories, +Topology 45 (2006), no. 2, 209–259, doi: 10.1016/j.top.2005.07.001. MR 2193334 +[8] +, C∗-Algebras over topological spaces: the bootstrap class, M¨unster J. Math. 2 (2009), +215–252. MR 2545613 +[9] +, Homological algebra in bivariant K-theory and other triangulated categories. I, Tri- +angulated categories (Thorsten Holm, Peter Jørgensen, and Rapha¨el Rouqier, eds.), London +Math. Soc. Lecture Note Ser., vol. 375, Cambridge Univ. Press, Cambridge, 2010, pp. 236–289. +MR 2681710 + + +34 +RALF MEYER AND RYSZARD NEST + +[10] Amnon Neeman, Triangulated categories, Annals of Mathematics Studies, vol. 148, Princeton +University Press, Princeton, NJ, 2001. MR 1812507 +[11] Gunnar Restorff, Classification of Cuntz–Krieger algebras up to stable isomorphism, J. Reine +Angew. Math. 598 (2006), 185–210, doi: 10.1515/CRELLE.2006.074. MR 2270572 +[12] +, Classification of Non-Simple C∗-Algebras, Ph.D. Thesis, Københavns Universitet, +2008, http://www.math.ku.dk/~restorff/papers/afhandling_med_ISBN.pdf. +[13] Mikael Rørdam, Classification of extensions of certain C∗-algebras by their six term exact +sequences in K-theory, Math. Ann. 308 (1997), no. 1, 93–117, doi: 10.1007/s002080050067. +MR 1446202 +[14] Steven Vickers, Topology via logic, Cambridge Tracts in Theoretical Computer Science, vol. 5, +Cambridge University Press, Cambridge, 1989. MR 1002193 + +Mathematisches Institut and Courant Research Centre “Higher Order Structures”, +Georg-August Universit¨at G¨ottingen, Bunsenstraße 3–5, 37073 G¨ottingen, Germany +E-mail address: rameyer@uni-math.gwdg.de + +Københavns Universitets Institut for Matematiske Fag, Universitetsparken 5, 2100 +København, Denmark +E-mail address: rnest@math.ku.dk + + diff --git a/papers/references/Loomis2018.pdf b/papers/references/Loomis2018.pdf new file mode 100644 index 00000000..7fb1bef8 --- /dev/null +++ b/papers/references/Loomis2018.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9d0c3dd63c265a76b970b47a8ee6a08e9302270feb78d39e30f5996f0bf9079f +size 140066 diff --git a/papers/references/Loomis2018.txt b/papers/references/Loomis2018.txt new file mode 100644 index 00000000..7041c4a7 --- /dev/null +++ b/papers/references/Loomis2018.txt @@ -0,0 +1,1035 @@ +arXiv:1709.00064v2 [gr-qc] 25 Sep 2017 + +September 2017 + +Suppression of non-manifold-like sets +in the causal set path integral + +S. P. Loomis∗ and S. Carlip† + +Department of Physics +University of California +Davis, CA 95616 +USA + +Abstract + +While it is possible to build causal sets that approximate spacetime +manifolds, most causal sets are not at all manifold-like. We show that a +Lorentzian path integral with the Einstein-Hilbert action has a phase +in which one large class of non-manifold-like causal sets is strongly +suppressed, and suggest a direction for generalization to other classes. +While we cannot yet show our argument holds for all non-manifold-like +sets, our results make it plausible that the path integral might lead to +emergent manifold-like behavior with no need for further conditions. + +∗email: sloomis@ucdavis.edu +†email: carlip@physics.ucdavis.edu + + +1. +Introduction + +The causal set program offers a simple, elegant picture of spacetime as a discrete set of points, +characterized solely by their causal relations. For all its elegance, though, causal set theory has +a potentially fatal flaw. +We know how to construct causal sets that approximate spacetime +manifolds, by starting with a manifold and extracting a Poisson “sprinkling” of points. But such +manifold-like sets are highly atypical; almost all causal sets do not look like any manifold at all. +If causal sets are fundamental, and manifold-like behavior is emergent, a dynamical process must +somehow suppress almost all typical causal sets, leaving only the rare manifold-like ones. Finding +such a process—especially one that has not been artificially constructed merely to achieve this +goal—is not easy. +In this paper, we show that the ordinary path integral with the causal set version of the +(Lorentzian) Einstein-Hilbert action has a phase in which one large class of non-manifold-like +causal sets is strongly suppressed. The class for which we can rigorously show this suppression, +the two-level orders, is itself not “typical”—we certainly do not claim to show that all non- +manifold-like sets are suppressed. But the two-level orders form a fairly large class, one much +larger than the class of manifold-like causal sets. As we discuss in the conclusion, there are +also hints that our methods may extend to more general classes. Our results thus make it more +plausible that the ordinary path integral, with no additional assumptions, may be enough to lead +to emergent manifold-like behavior. +A numerical analysis of two-dimensional causal sets has shown a similar transition between +a phase dominated by non-manifold-like causal sets and one dominated by manifold-like sets [1]. +In one way, that result is stronger than ours, since it accounts for all non-manifold-like sets. On +the other hand, our results are analytic, hold in any dimension, and use the Lorentzian path +integral rather than analytically continuing to Riemannian signature. + +2. +Non-manifold-like causal sets + +The procedure for constructing a manifold-like causal set is well understood [2]. One starts +with a finite-volume region of a manifold with a Lorentzian metric, “sprinkles” points randomly +by a Poisson process, determines the causal relations among these points from the causal structure +of the manifold, and then “forgets” the manifold, keeping only the points and their causal +relations. For a dense enough sprinkling of points, the resulting causal set retains the fundamental +properties of the original manifold: the Alexandrov neighborhoods determine the topology, the +causal relations determine the conformal class of the metric, and the density of points determines +the conformal factor [3,4]. +But such manifold-like causal sets are highly atypical. The “typical” causal set is a Kleitman- +Rothschild (KR) order, a three-level causal set with approximately n/4 points in the “bottom” +and “top” layers and n/2 points in the “middle” layer [5]. In fact, as n → ∞, the proportion of +n-element causal sets that are KR orders goes to one. +Many other non-manifold-like causal sets also occur frequently. There is, in fact, a hierarchy +of classes of non-manifold-like causal sets [6–9]. Each class is characterized by a parameter p + +1 + + +that is the proportion of possible relations that are actualized, and is dominated by causal sets +with a particular number of levels. The dependence of the size of the class on p is not smooth, +but is described by an piecewise continuous function with infinitely many “phase transitions” +characterized by either the creation of new layers or changes in the relative sizes of the layers. +The intricacies of these classes are beyond the scope of this paper—see [8] for details—but it is +sufficient to point out that the class of non-manifold-like causal sets is dominated by three-level +orders, primarily the KR orders, followed by two-level orders and then four-level orders. +In this paper we will focus on the simplest case of two-level orders. Though these are not +as dominant as the three-level orders, they still form a significant part of the collection of non- +manifold like causal sets. + +3. +Causal set path integrals + +To define a path integral for causal sets, we need two ingredients: an appropriate generaliza- +tion of the Einstein-Hilbert action and a discrete version of an integration measure. The action +we shall use, the Benincasa-Dowker action, was introduced in [10]. For a causal set C with n +elements, it takes the general form [11,12] + +1 +ℏS(C) = µ + +� + +n + + +kmax +� + +k=0 +λkNk + +� + +(3.1) + +where µ and λk are appropriately chosen parameters and Nk denotes the number of pairs of +elements {x, y} ⊂ C such that the cardinality of the set {z ∈ C : x ≺ z ≺ y} is equal to k. The +upper limit kmax can be finite or infinite, though it has a lower bound of ⌊2 + d + +2⌋, where d is the +target spacetime dimension. +Eq. (3.1) replicates the Einstein-Hilbert action in the following sense. Suppose we construct +a causal set by Poisson sprinkling points into a manifold of the target dimension. Then for a +high enough sprinkling density and the correct choices of µ and λk, S(C) is equal to the Einstein- +Hilbert action on average. The specific definitions of µ and λk are complicated, but for d = 4 +and kmax = 3 we have + +1 +ℏS(C) = +� l + +lp + +�2 +(n − N0 + 9N1 − 16N2 + 8N3) +(3.2) + +where lp is the Planck length and l is a length scale determined by the sprinkling density of +events into the spacetime. +For our “integration measure” we shall simply sum over causal sets. As in causal dynamical +triangulations [13], we should perhaps include a combinatorial weight to avoid overcounting +causal sets with special symmetries, but that will not affect our conclusions. The Lorentzian +partition function over any particular class C of causal sets is then + +Z[µ, λ0] = +� + +C∈C +exp +� i + +ℏS(C) +� += +� + +C∈C +exp + +� + +iµ + +� + +n + + +kmax +� + +k=0 +λkNk + +�� + +(3.3) + +2 + + +We will be interested in the large n behavior of this quantity; for a manifold-like causal set with +a fixed sprinkling density, this is the large volume limit. + +4. +Suppression of two-level orders + +For this paper we focus on two-level orders, that is, causal sets C of size n such that there +are no three distinct elements x, y, z ∈ C satisfying x ≺ y ≺ z. This means that Nk = 0 for +k > 0. Intuitively, such sets have only two “moments of time,” and clearly do not resemble +manifolds. As we have mentioned, while they are less common than the three-level KR orders, +two-level orders are still much more common than manifold-like causal sets, and they threaten +to dominate the path integral. +For any n-element causal set, N0 can be no larger than Nmax = n(n−1) + +2 +. We classify such sets +by the proportion 0 ≤ p ≤ 1 of relations, given by N0 = pNmax. For fixed n, p is a discrete +parameter, but in the limit of large n we can approximate it as continuous. The utility of this +classification is that the Benincasa-Dowker action is constant over the class of two-level sets with +a fixed p. Denoting such a class by Cp,n, we can write the partition function over two-level orders +of size n as + +Z[µ, λ0] = +� +dp |Cp,n|eiS(p)/ℏ = eiµn +� 1 + +0 +dp |Cp,n| exp +�1 + +2iµλ0pn2 + o(n2) +� +(4.1) + +where |Cp,n| is the cardinality of the class Cp,n. Here we have written Nmax = 1 + +2n2 + o(n2), where +o(n2) denotes terms subleading to n2, which will be negligible in the large n limit. +To calculate |Cp,n| we consider a decomposition into classes Cq,p,n where we put qn of the +elements in the “top” level and (1−q)n in the “bottom” level. Let us denote α = q(1−q), where +α ≤ 1 + +4 since 0 ≤ q ≤ 1. From the structure of the system, there can be at most αn2 relations—the +maximum occurs when every “bottom” element is related to every “top” element—so from the +definition of p, we have α ≥ 1 + +2p. This in turn implies that p ≤ 1 + +2. +The number of ways to choose pNmax = 1 + +2pn(n − 1) pairs from the possible αn2 relations is + +|Cq,p,n| = +� +αn2 + +1 +2pn(n − 1) + +� +(4.2) + +With both arguments large, we can expand the binomial as + +ln |Cq,p,n| =αn2 ln(αn2) − 1 + +2pn2 ln +�1 + +2pn2 +� +− +� +α − 1 + +2p +� +n2 ln +�� +α − 1 + +2p +� +n2 +� ++ o(n2) + += +� +α ln α − 1 + +2p ln +�1 + +2p +� +− +� +α − 1 + +2p +� +ln +� +α − 1 + +2p +�� +n2 + o(n2) +(4.3) + +For 1 + +2p ≤ α ≤ 1 + +4, this is is a monotonically increasing function of α. This means that |Cq,p,n| is +maximized for q = 1 + +2. Now, |Cp,n| is bounded by +���C 1 + +2,p,n +��� ≤ |Cp,n| ≤ +� + +q +|Cq,p,n| +(4.4) + +3 + + +In the large n limit, the upper bound is dominated by the maximal value of q, so + +ln |Cp,n| = ln |C 1 + +2 ,p,n| + o(n2) = 1 + +4h(2p)n2 + o(n2) +� +p ≤ 1 + +2 + +� +(4.5) + +where +h(x) = −x ln x − (1 − x) ln(1 − x) +(4.6) + +is the entropy function. (As we saw above, p ≤ 1 + +2 for two-level sets, so |Cp,n| = 0 for p > 1 + +2.) +Using (4.5), we can write the partition function as + +Z[µ, λ0] = eiµn +� 1/2 + +0 +dp exp +�1 + +2iµλ0pn2 + 1 + +4h(2p)n2 + o(n2) +� +(4.7) + +To simplify notation, we define + +− µλ0 + +2 += β, +2p = x +(4.8) + +Note that 0 ≤ x ≤ 1 and that, from (3.2), β > 0. The exponent in (4.7) is then n2 + +4 E(x), with + +E(x) = −2iβx + h(x) +(4.9) + +We will evaluate the integral by the method of steepest descents.∗ Here we sketch the method +and results; details are given in the appendix. We first find the saddle point: + +E′(x) = 0 = −2iβ − ln x + ln(1 − x) +(4.10) + +⇒ x0 = +e−iβ + +2 cos β = 1 + +2(1 − i tan β), +1 − x0 = +eiβ + +2 cos β = 1 + +2(1 + i tan β) + +The second derivative at x = x0 is + +E′′(x0) = − 1 + +x0 +− +1 + +1 − x0 += −4 cos2 β +(4.11) + +so the direction of steepest descent is x − x0 real. At the saddle point, + +h(x0) = − e−iβ + +2 cos β ln +� e−iβ + +2 cos β + +� +− +eiβ + +2 cos β ln +� +eiβ + +2 cos β + +� += β tan β + ln(2 cos β) + +E(x0) = −2iβx0 + h(x0) = −iβ + ln(2 cos β) +(4.12) + +Remembering that the exponent is n2 + +4 E(x), we have a saddle point contribution of + +Z[µ, λ0] ∼ eiµn + +n + +� +π + +2|E′′(x0)| exp +�n2 + +4 E(x0) +� += +�π + +8 +eiµn + +n cos β exp +�n2 + +4 [−iβ + ln(2 cos β)] +� +(4.13) + +If | cos β| < 1 + +2, the real part of the exponent is negative, and the path integral is exponentially +suppressed. + +∗An earlier attempt to determine the integral in a quadratic approximation failed; we thank Lisa Glaser for +pointing out an algebraic error that invalidated our first approach. + +4 + + +• +• + +• +x− + +0 +1 + +C− +1 +C− +2 + +C− +3 + +Figure 1: Deformed contour through the saddle point at x0 with tan β > 0 + +This is not quite the whole story. The method of steepest descent requires a contour defor- +mation, and we must check that the rest of the contour does not spoil the result. For tan β > 0, +the saddle point is in the lower half plane, and the contour is shown in figure 1. +We show +in the appendix that the remaining pieces of the contour, C− +1 and C− +2 , are also exponentially +suppressed. If, on the other hand, tan β < 0, we must deform the contour into the upper half +plane, and the remaining pieces are not suppressed. We thus conclude that the path integral for +two-level orders is exponentially suppressed at large volume provided that + +tan +� +−µλ0 + +2 + +� +> 0 +and +����cos µλ0 + +2 + +���� < 1 + +2 +⇒ tan +� +−µλ0 + +2 + +� +> +√ + +3 +(4.14) + +We can also carry the analysis one step further. The saddle point approximation (4.13) is +not exact, and one might worry about the higher order terms in the exponent. In the appendix, +we give a rigorous bound: exponential suppression is guaranteed provided that + +tan +� +−µλ0 + +2 + +� +> +�27 + +4 e−1/2 − 1 +�1/2 +≈ 1.759 +(4.15) + +This gives a minimum value of |µλ0| ≈ 2.108, or a scale ℓ ≈ 1.452ℓp in (3.2). + +5. +Discussion + +The program we have described can be summarized as follows: + +1. Identify a class of causal sets that can be divided into subclasses characterized by some +parameters pi such that the action is constant over each subclass. + +2. Count how large each subclass is, to leading order in the size n of the set, as a function of +the parameters pi. + +3. Analytically evaluate the partition function as an integral over pi, and study how it depends +on the parameters µ and λi in the action. + +We have carried this out for a particularly simple case, in which the division into easily +countable subclasses was fairly straightforward. +But there are hints that our results can be + +5 + + +generalized. Once we move beyond two-level orders, the action (3.2) will include contributions +from N1, N2, and N3, greatly complicating the counting. But for sets with only a few levels, +these contributions may be strongly suppressed. +Consider, for example, a KR order, which has approximately n/4 points in a “bottom” level, +n/2 in a “middle” level, and n/4 in a “top” level.† Pick a “bottom” point x and a “top” point +y. Typically, x will link to approximately n/4 points in the middle level. Imagine coloring these +points red, and the remaining points blue. For {x, y} to contribute to N1, y must then link to +exactly one red point in the middle level, along with approximately n/4 blue points. It is easy +to see that the probability of such a pattern goes as a polynomial in n times 2−n/2. The same is +true for contributions to N2 (for which y must link to exactly two red points) and N3 (for which +y must link to exactly three red points). Similar arguments should hold whenever the number +of levels is small. +This suppression should reduce the analysis of KR orders, and perhaps similar few-level +sets, to the form we have already considered, in which only N0 is important. This is still a +preliminary argument, of course. The N0 combinatorics will be different for different orders, +and one must check that the “atypical” few-level causal sets—three-level sets with different +distributions of points or relations from the KR orders, for instance—remain subdominant. Here +the combinatoric results of [8] may prove useful, but much more work is needed. +There are +also subtleties involving the difference between labeled and unlabeled causal sets that require +careful attention [9]. Numerical exploration of distributions of causal sets and relations may shed +additional light on these problems. + +Acknowledgments + +We are very grateful to Lisa Glaser for pointing out a crucial error in an early version of this +work. We also thank David Rideout for helpful conversations. This work was supported in part +by U.S. Department of Energy grant DE-FG02-91ER40674. + +Appendix. +Steepest descent details + +In this appendix we describe some of the details involved in the steepest descent calculation +of section 4. + +Contours + +The integral (4.7) is over the interval 0 < x < 1. For the method of steepest descent, we +must first deform the contour to go through the saddle point in the direction of steepest descent. +The saddle point is x0 = 1 + +2(1 − i tan β) and the direction of steepest descent is x − x0 real, so the +contours are those of figure 2, where the lower branch is applicable for tan β > 0 and the upper +for tan β < 0. + +†More precisely [5], a KR order has between n/4 − n1/2 ln n and n/4 + n1/2 ln n points in the bottom and top +levels, and between n/2 − ln n and n/2 + ln n points in the middle level. + +6 + + +• +• + +• +x− + +0 +1 + +C− +1 +C− +2 + +C− +3 + +• +x+ + +C+ +1 +C+ +2 + +C+ +3 + +Figure 2: Deformed contours through the saddle point at x0 + +Let us first exclude the contour in the upper half plane. Consider C+ +1 . We can write + +x = +iw, +0 < w < 1 + +2| tan β| +(A.1) + +from which, with the branch cuts shown in figure 2, + +ln x = πi + +2 + ln w, +ln(1 − x) = ln +√ + +1 + w2 − i tan−1 w +(A.2) + +with the inverse tangent lying between 0 and π + +2. Hence + +h(x) = −x ln x − (1 − x) ln(1 − x) + += −iw +�πi + +2 + ln w +� +− (1 − iw) +� +ln +√ + +1 + w2 − i tan−1 w +� + += π + +2 w + w tan−1 w − ln +√ + +1 + w2 + imaginary part +(A.3) + +For ℑx > 0, the contribution from the term −2iβx in E(x) is positive, and + +ℜ E = +�π + +2 + tan−1 w + 2β +� +w − ln +√ + +1 + w2 +(A.4) + +For positive real w, this is always positive, so the integral acquires an exponentially large con- +tribution from C+ +1 . This rules out the saddle point approximation for this contour. +Next consider the contour in the lower half plane. On C− +1 , we can write + +x = −iw, +0 ≤ w ≤ 1 + +2 tan β + +ln(−iw) = −πi + +2 + ln w, +ln(1 + iw) = ln +√ + +1 + w2 + i tan−1 w +(A.5) + +7 + + +with the inverse tangent again lying between 0 and π + +2. Then + +h(x) = iw +� +−πi + +2 + ln w +� +− (1 + iw) +� +ln +√ + +1 + w2 + i tan−1 w +� + += π + +2 w + w tan−1 w − ln +√ + +1 + w2 + imaginary part +(A.6) + +and thus +ℜ E = +�π + +2 + tan−1 w − 2β +� +w − ln +√ + +1 + w2 +(A.7) + +For β > π + +2, this is always negative, and the contribution from C− +1 is exponentially suppressed. +For 0 < β < π + +2, the requirement that | cos β| < 1 + +2 limits us to the range π + +3 < β < π + +2. To proceed, +let us determine the maximum value of ℜ E in this range. +Note first that at w = 0, ℜ E = 0 and the derivative + +d(ℜ E) + +dw += π + +2 + tan−1 w − 2β +(A.8) + +is negative, so ℜ E < 0 for small w. The turning point occurs at + +π +2 + tan−1 w − 2β = 0 ⇒ w = − cot 2β = 1 + +2(tan β − cot β) +(A.9) + +For 1 + +2(tan β − cot β) < w < 1 + +2 tan β, ℜ E is increasing, so its maximum in this range will occur +at the endpoint w = 1 + +2 tan β. At that maximum, + +ℜ E = 1 + +2 + +�π + +2 + tan−1 +�1 + +2 tan β +� +− 2β +� +tan β − ln + +� + +1 + 1 + +4 tan2 β +(A.10) + +Treating this quantity as a function of β and using Mathematica [14] to determine its zeros, we +find that it is negative for .9474 < β < π + +2, an interval that includes the full range of interest. +Hence ℜ E(w) < 0 for any β in the range π + +3 < β < π + +2, and the contribution of the contour C− +1 is +again exponentially suppressed. +The contour C− +2 is basically a reflection, and gives the identical suppression. Let + +x = 1 − iv, +0 ≤ v ≤ 1 + +2 tan β +(A.11) + +Then + +h(x) = −(1 − iv) +� +ln +√ + +1 + v2 − i tan−1 v +� +− iv +�πi + +2 + ln v +� + += π + +2 v + v tan−1 v − ln +√ + +1 + v2 + imaginary part +(A.12) + +and +ℜ E = +�π + +2 + tan−1 v − 2β +� +v − ln +√ + +1 + v2 +(A.13) + +8 + + +which exactly matches (A.7). This match is not accidental; it follows from the fact that + +ℜh(1 − iv) = ℜh(iv) = ℜh(−iv) + +as long as we stay on the same branch of the logarithm. + +Error estimates + +The integral (4.13) is based on a quadratic approximation to E(x). In this case, we can also +get control over the errors. Let x = 1 + +2(1 − u). It is then easy to check that for n ≥ 2, + +dnh +dun = −1 + +2 +(n − 2)! +(1 − u)n−1 − (−1)n + +2 +(n − 2)! +(1 + u)n−1 +(A.14) + +Now expand E(x) around x0. Since u0 = i tan β is imaginary, the two terms in (A.14) evaluated +at x0 are complex conjugates; the odd derivatives are imaginary, while the even derivatives are +real. The Taylor expansion for E(x) around x0, with x − x0 real, is then + +ℜE(x) = ℜE(x0) − + +∞ +� + +n=1 + +22n + +2n(2n − 1)[cos2n−1β][cos(2n − 1)β] (x − x0)2n +(A.15) + +where (4.10) has been used to evaluate ℜ(1 − u0)−(2n−1). Hence +��ℜ(E(x) − E(x0) + 2 cos2 β(x − x0)2) +�� + +≤ + +∞ +� + +n=2 + +22n + +2n(2n − 1)| cos2n−1β|| cos(2n − 1)β| (x − x0)2n + +≤ + +∞ +� + +n=2 + +22n + +2n(2n − 1) + +�1 + +2 + +�4n−1 += + +∞ +� + +n=2 + +2−2n + +n(2n − 1) +(A.16) + +using the facts that | cos β| ≤ 1 + +2 and |x − x0| ≤ 1 + +2. The sum evaluates to + +3 +2 ln 3 + +2 + 1 + +2 ln 1 + +2 − 1 + +4 ≈ 0.0116 + +We can thus state, for instance, that on the line 0 < ℜx < 1, ℑx = − i + +2 tan β with tan β > 0—that +is, the line through the saddle point x0—the exponent E(x) is negative as long as + +| cos β| < 2 · 3−3/2e−1/4 ≈ 0.4942 +(A.17) + +which in turn yields (4.15). We do not know whether this is a sharp limit. + +References + +[1] L. Glaser and S. Surya, Class. Quant. Grav 33 (2016) 065003, arXiv:1410.8775. + +9 + + +[2] L. Bombelli, J. Lee, D. Meyer, and R. Sorkin, Phys. Rev. Lett. 59 (1987) 521. + +[3] L. Bombelli and D. A. Meyer, Phys. Lett. A141 (1989) 226. + +[4] S. Major, D. Rideout, and S. Surya, J. Math. Phys. 48 (2007) 032501, arXiv:gr-qc/0604124. + +[5] D. J. Kleitman and B. L. Rothschild, Trans. Amer. Math. Soc. 205 (1975) 205. + +[6] D. Dhar, J. Math. Phys. 19(8) (1978) 1711. + +[7] D J. Kleitman and B. L. Rothschild, Physica 96A (1979) 254. + +[8] H J. Pr¨omel, A. Steger, and A. Taraz, J. Combin. Theory, Series A 94 (2001) 230. + +[9] J. Henson, D. P. Rideout, R. D. Sorkin, and S. Surya, arXiv:1504.05902. + +[10] D. M. T. Benincasa and F. Dowker, Phys. Rev. Lett. 104 (2010) 181301, arXiv:1001.2725. + +[11] F. Dowker and L. Glaser, Class. Quant. Grav 30 (2013) 195016, arXiv:1305.2588. + +[12] L. Glaser, Class. Quant. Grav 31 (2014) 095007, arXiv:1311.1701. + +[13] J. Ambjørn, J. Jurkiewicz, and R. Loll, Nucl. Phys. B610 (2001) 347, arXiv:hep-th/0105267. + +[14] Wolfram Research, Inc., Mathematica, Version 11.1, Champaign, IL (2017). + +10 + +