A more general method to classify up to equivariant KK-equivalence

Using a homological invariant together with an obstruction class in a certain Ext^2-group, we may classify objects in triangulated categories that have projective resolutions of length two. This invariant gives strong classification results for actions of the circle group on C*-algebras, C*-algebras over finite unique path spaces, and graph C*-algebras with finitely many ideals.


Introduction
The C * -algebra classification program aims at classifying certain C * -algebras up to isomorphism by suitable invariants.Such a classification usually has two steps.First, an isomorphism between the invariants is lifted to an equivalence in a suitable equivariant KK-theory; then the latter is lifted to an isomorphism.These two steps are quite different in nature.The first is mainly algebraic topology, the second mainly analysis.This article deals with the first step of getting equivariant KK-equivalences from isomorphisms on suitable invariants.
The invariants used previously are homological functors -variants of K-theory.There are, however, many situations where no homological invariant is known that is sufficiently fine to detect KK-equivalences.This article introduces a more complex invariant with two layers: the primary invariant is a homological functor as usual, the secondary is a certain obstruction class, which lives in an Ext 2 -group constructed from the primary invariant.We took this idea from Wolbert [35].It goes back further to Bousfield [5].
Our two-layer invariants are complete invariants up to KK-equivalence in several new cases and shed light on previous classification results for non-simple Cuntz-Krieger algebras and graph algebras.We explain how to classify arbitrary objects in the bootstrap class in KK T , where T is the circle group, and in KK(X) for a finite unique path space X (see Definition 4.2).The latter result is far more general than previous ones in [3,26].Furthermore, we deduce a classification theorem up to stable isomorphism for purely infinite graph C * -algebras with finitely many ideals; this contains the class of real-rank-zero Cuntz-Krieger algebras classified by Restorff [31].Our approach has the additional advantage that the resulting classification result is strong, that is, every isomorphism on the level of invariants lifts to an isomorphism of C * -algebras; this also leads to a classification theorem up to actual isomorphism for the class of unital graph C * -algebras as above.
Our method is based on homological algebra in triangulated categories, see [21,25].This starts with a homological invariant on a triangulated category, which defines a homological ideal as its kernel on morphisms.The general theory gives projective resolutions, derived functors, and a Universal Coefficient Theorem for objects with a projective resolution of length 1; this implies that a certain universal homological invariant -in practice, this is often the one we started from -is a 2010 Mathematics Subject Classification.Primary 46L35; Secondary 18E30, 19K35.The first author was partially supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92).
Even if X is not a unique path space, our classification theorem applies to objects in the bootstrap class in KK(X) that have projective resolutions of length 2. We show that this is the case for graph C * -algebras with finitely many ideals.Furthermore, we compute the obstruction class of a graph algebra from the Pimsner-Voiculescu type sequences that compute the K-theory groups of its ideals.Hence our complete invariant may be computed effectively in this case.We get a strong classification functor up to stable isomorphism for purely infinite graph C * -algebras with finitely many ideals; strong classification means that every isomorphism on the invariants lifts to a stable isomorphism.This is the first strong classification result -even for the class of purely infinite Cuntz-Krieger algebras -without the assumption of a specific ideal structure.The invariant and its computation are described in more detail in Section 5.
Our abstract setup should also work in many other situations.One of them is connective K-theory, regarded as an invariant on connective E-theory.We refer to Andreas Thom's thesis [34] for details.See [9] for applications of connective K-theory to C * -algebras.Another instance is Kasparov's KK-theory for C * -algebras over a zero-dimensional compact metrisable space X.Here the K-theory of the total algebra has a natural module structure over the ring of locally constant functions C(X, Z).This ring has global dimension 2 by [12,Examples 2.5(b)].
For C * -algebras over the unit interval and filtrated K-theory as the invariant, the relevant Abelian category has dimension 2 once again.So far, we cannot treat this example, however, because there are not enough projective objects in this case.
1.1.Outline.The structure of this article is as follows.Section 2 develops the general theory of obstruction classes.Section 3 applies it to circle actions on C * -algebras, Section 4 to C * -algebras over unique path spaces, and Section 5 to graph C * -algebras; this includes a return to general triangulated categories in order to compute obstruction classes for objects of a specific type.1.2.Acknowledgement.We thank James Gabe and Rune Johansen for pointing out the references [12] and [6], respectively; Eusebio Gardella and N. Christopher Phillips for helpful discussions on circle actions with the Rokhlin property, and Gunnar Restorff for helpful discussions on the classification of graph C * -algebras.

Lifting two-dimensional objects
Throughout the article, we will freely use the language of homological algebra in triangulated categories introduced in [25].
Let T be a triangulated category with at least countable direct sums (so that idempotent morphisms split).Let I be a stable homological ideal that is compatible with countable direct sums and has enough projective objects.Let F : T → A be the universal I-exact stable homological functor.By [25, Theorem 57], the category A has enough projective objects, the adjoint functor F ⊢ of F is defined on all projective objects of A, and F • F ⊢ (A) ∼ = A for all projective objects A of A. Let P I ⊆ T denote the localising subcategory generated by the I-projective objects.[21,Theorem 3.22] implies that Â ∈∈ P I if and only if T( Â, B) = 0 for all I-contractible objects B ∈∈ T; we write ∈∈ for objects of a category.

Definition 2.1. A lifting of
We often drop α from the notation and call Â a lifting of A.

Proposition 2.2. Let A ∈∈ A have cohomological dimension 1.
Then A has a lifting, and any two liftings are equivalent.
0 and hence T( Â, B) = 0 by the long exact sequence for T(␣, B).Hence Â is a lifting of A.
Let Â1 and Â2 be liftings of A. This includes a choice of isomorphisms F ( Â1 ) ∼ = A and F ( Â2 ) ∼ = A. The Universal Coefficient Theorem [25, Theorem 66] applies to T( Â1 , Â2 ).Hence there is f ∈ T( Â1 , Â2 ) that lifts the identity map on A when we identify F ( Â1 ) ∼ = A and F ( Â2 ) ∼ = A. Since f is an I-equivalence, its cone B is I-contractible.Thus T * ( Âi , B) = 0 for i = 1, 2, and this implies T * (B, B) = 0 and hence B = 0 by the long exact sequence.Thus f is invertible.
The equivalence between two liftings in Proposition 2.2 is not canonical, and the lifting is not natural, unlike for projective objects.The Universal Coefficient Theorem [25,Theorem 66] only shows that any arrow A 1 → A 2 in A between objects of cohomological dimension 1 lifts to an arrow in T. But this lifting is only unique up to Ext ).With parity assumptions as in Section 2.1, there is a canonical lifting for any arrow A 1 → A 2 : lift its even and odd parts separately and then take the direct sum.This shows that the UCT short exact sequence splits under parity assumptions.This splitting is not natural, however.Proposition 2.2 implies that isomorphism classes of objects in A of cohomological dimension 1 correspond bijectively to isomorphism classes of objects A in P I with F (A) of cohomological dimension 1.This is used in [3,19,26] and other classification results.It may, however, be very hard to find computable invariants F for which all objects in its image have cohomological dimension 1.
Hence the inclusion map ΩA ֒→ P 0 lifts to some f ∈ T(D, P0 ), which is I-monic.
The mapping cone Â of f belongs to P I by construction, and has F ( Â) ∼ = P 0 /ΩA ∼ = A by the short exact sequence (2.4), so it is a lifting of A.
[26, Theorem 4.10] shows that liftings of objects of cohomological dimension two cannot be unique in general.We may, however, classify liftings up to equivalence:

Theorem 2.6. Let A ∈∈ A have cohomological dimension 2. The set of equivalence classes of liftings of A is in bijection with Ext 2 (A, A[−1]).
Proof.Fix a length-two projective resolution of A as in (2.4) and a lifting Â of A, which exists by Lemma 2.3.Let P0 := F ⊢ (P 0 ).The map P 0 → A in A is adjoint to a map P0 → Â in T. We may complete this to an exact triangle Since ΩA has cohomological dimension 1, its lifting D is unique up to isomorphism by Proposition 2.2.The exact triangle (2.7) shows that Â is the cone of ϕ.So any other lifting Â′ must be the cone of some arrow ϕ ′ : D → P0 that induces the inclusion map F (D) → P 0 .Conversely, if ϕ ′ : D → P0 lifts the inclusion map ΩA → P 0 , then its cone is a lifting of A.
Let Â and Â′ be the liftings associated to ϕ and ϕ ′ .We claim that an isomorphism α : Â → Â′ that induces the identity map on F ( Â′ ) ∼ = A ∼ = F ( Â) may be embedded in a morphism of triangles (2.8) The assumption that α induces the identity map on A means that the right square commutes.This allows to find an arrow ψ : D → D to give a triangle morphism, by an axiom of triangulated categories.The arrow ψ induces the identity map on F (D) = ΩA because the map F (D) → P 0 is injective.Thus ϕ ′ • ψ = ϕ for some ψ : D → D that induces the identity map on F (D).
Conversely, let ϕ ′ • ψ = ϕ for some ψ : D → D that induces the identity map on F (D).This means that the left square in (2.8) commutes.Embed this square in a triangle morphism to construct α : Â → Â′ .Since ψ induces the identity map on F (D), it is invertible.Hence α is also invertible by the Five Lemma for exact triangles.Summing up, Â and Â′ are equivalent liftings if and only if there is By the Universal Coefficient Theorem, the possible choices for ψ − id D and ϕ ′ − ϕ lie in Ext 1 (ΩA, ΩA[−1]) and Ext 1 (ΩA, P 0 [−1]), respectively.The short exact sequence ΩA P 0 ։ A induces a long exact sequence The naturality of the UCT sequence implies that composition with ψ acts on Ext 1 (ΩA, P 0 [−1]) by adding j( ψ), where ψ ∈ Ext 1 (ΩA, ΩA[−1]) is the class associated to ψ − id D .Hence Â and Â′ are equivalent if and only if ϕ ′ − ϕ is mapped to 0 in Ext 1 (ΩA, A[−1]), and any element in Ext 1 (ΩA, A[−1]) arises from some ϕ ′ .Since P 0 is projective, another long exact sequence implies Thus Â and Â′ are equivalent if and only if ϕ ′ − ϕ is mapped to 0 in Ext 2 (A, A[−1]), and any element in Ext 2 (A, A[−1]) comes from some ϕ ′ .
We claim that the map sending Â′ to the image of ϕ ′ − ϕ in Ext There is a cohomological spectral sequence associated to the ideal I, called ABC spectral sequence in [21] after Adams, Brinkmann and Christensen.It is discussed in great detail in [21,Section 4].The relative obstruction class δ( Â2 , Â1 ) is related to the boundary map on the second page of the ABC spectral sequence for T( Â1 , Â2 ).The E 2 -term in this spectral sequence is ) for p ≥ 0, q ∈ Z.By assumption, E p,q 2 = 0 for p = 0, 1, 2. Hence E p,q k = 0 for p = 0, 1, 2 and k ≥ 3 as well.Since the boundary map d k on E k has bidegree (k, 1 − k), we get d k = 0 for k ≥ 3, and the only part of d 2 that may be non-zero is d 0,q 2 : E 0,q 2 → E 2,q−1 2 . Hence As a consequence, ϕ ∈ Hom(A, A) lifts to T( Â1 , Â2 ) if and only if d 0,0 2 (ϕ) = 0.In particular, Â1 and Â2 are equivalent liftings if and only if id A ∈ Hom(A, A) lifts to T( Â1 , Â2 ), if and only if vanishes.Thus both conditions d 0,0 2 (id A ) = 0 and δ( Â1 , Â2 ) = 0 are necessary and sufficient for an equivalence of liftings.This suggests the following lemma: Lemma 2.11.d 0,0 2 (id A ) = δ( Â1 , Â2 ) = −δ( Â2 , Â1 ).Proof.The cohomological spectral sequence for T( Â1 , Â2 ) in [21, Section 4] is constructed using a phantom tower for Â1 .We implicitly already constructed such a phantom tower when lifting A to Â1 .In the notation above, it looks as follows: (2.12) The circled arrows have degree −1.(The conventions about the degrees of the maps in the phantom tower are different in [21].)Here Pj and ∂j are the unique liftings of the projective objects P j and the boundary maps ∂ j in (2.4) for j = 0, 1, 2, and ϕ 1 is the map with cone Â1 used in the arguments above.The triangles involving ι n are I-exact, and the other triangles commute.This together with the I-projectivity of the objects Pn means that (2.12) is a phantom tower.The relevant cohomological spectral sequence is constructed by applying the cohomological functor T(␣, Â2 ) to the phantom tower for Â1 in (2.12).The boundary map d 2 on the second page maps , where We describe how d 2 acts on id The UCT exact sequence for T * (D, Â2 ) is the long exact sequence associated to the triangle P2 P1 D in (2.12).This exact sequence shows that . The map ϕ 2 − ϕ 1 induces the zero map F (D) → F ( P0 ) and hence corresponds to an element x in Ext 1 (ΩA, P 0 [−1]) by the UCT sequence.By definition, the obstruction class δ( Â2 , Â1 ) is the image of x under the map where the first map is induced by the projection P 0 → A. By the naturality of the UCT sequence, this maps Our description of equivalence classes of liftings is not yet a classification of objects in P I up to isomorphism.Two objects Â1 , Â2 ∈ P I are isomorphic if and only if there is an isomorphism F ( Â1 ) → F ( Â2 ) that lifts to T( Â1 , Â2 ).If F ( Â1 ) = F ( Â2 ) and δ( Â1 , Â2 ) = 0, then the identity map F ( Â1 ) → F ( Â2 ) does not lift; but there may be another isomorphism F ( Â1 ) ∼ = F ( Â2 ) that lifts to T( Â1 , Â2 ).This seems hard to decide given only F ( Âi ) and δ( Â1 , Â2 ) = 0.
2.1.Parity assumptions.We are going to impose an extra assumption on A that provides a canonical lifting for each object of A of cohomological dimension 2. This allows us to understand the action of automorphisms on obstruction classes and to classify objects of P I with length-2-projective resolutions up to isomorphism.

Definition 2.13. A stable Abelian category is called even if
is, any object of A is a direct sum of objects of even and odd parity, and the suspension automorphism on A shifts parity.
Example 2.14.Let A be the category of countable, Z/2-graded modules over a ring R. Then A is even, with A ± the subcategories of countable R-modules concentrated in even or odd degree, respectively.
If T = KK G for a compact group G and I is the kernel of morphisms of the functor K G * , then the Abelian approximation of T with respect to I is the category of countable, Z/2-graded modules over the representation ring of G. Hence A is even in this example.
Assume that A is even.Since the two subcategories A ± are orthogonal, we have Ext shows that there are unique liftings Â+ and Â− for A + and A − (up to equivalence).We call Â0 := Â+ ⊕ Â− ∈∈ T the canonical lifting of A and let δ( Â) := δ( Â, Â0 ) for any other lifting.This defines a canonical obstruction class in Ext 2 (A, A[−1]) for all liftings Â of A. A simple computation as in [35,Proposition 9] shows that, for f ∈ Hom(A, B), the element ) is given by the formula (2.15) There is an additive functor The following classification result generalises [5, Theorem 9.1] and [35,Theorem 11].
Theorem 2.17.Assume that A is even and has global dimension 2. Then the functor F δ is full and induces a bijection between isomorphism classes of objects Â in P I and isomorphism classes of objects in the category Aδ.Furthermore, every lift of an isomorphism in Aδ is an isomorphism in T.
Proof.The last claim in the theorem follows from a standard argument: if Â1 and Â2 belong to P I and if f ∈ T( Â1 , Â2 ) is an I-equivalence, then the mapping cone C f of f is both I-contractible and in The proof of Theorem 2.6 shows that every class in Ext 2 (A, A[−1]) appears as δ( Â) for some lifting Â of A. Hence F δ is essentially surjective.Since a morphism f ∈ Hom(A, B) lifts to a morphism Â → B if (and only if) d 0,0 2 (f ) = 0, (2.15) shows that the functor F δ is full.Hence F δ distinguishes isomorphism classes.

Kasparov theory for circle actions
Let G be a connected compact Lie group with torsion-free fundamental group.We will soon specialise to the circle group G = T, but some results hold more generally.Let T := KK G be the G-equivariant Kasparov theory.It has C * -algebras with a continuous G-action as objects and KK G 0 (A, B) as arrows from A to B. Its triangulated category structure is introduced in [22].
The representation ring separable because it is the K-theory of a separable C * -algebra.Let A be the category of countable, Z/2-graded R-modules.Let F := K G * : T → A be the equivariant K-theory functor.Let I be the kernel of F on morphisms.This is a stable homological ideal by definition.
[25, Theorem 72] says that F is the universal I-exact stable homological functor and that I has enough projective objects.More precisely, the adjoint F ⊢ maps the free rank-one module R to C with trivial G-action because There is a Morita-Rieffel equivalence Let n be the rank of the maximal torus in G and let W be the Weyl group of G.
, where the action of W comes from the canonical action on T .Even more, we have for some l with 0 ≤ l ≤ n; see for instance [33].This ring has cohomological dimension n + 1 because Z has cohomological dimension 1 and each independent variable adds 1 to the length of resolutions.
The cohomological dimension of R is 2 if and only if n = 1.The two groups G with n = 1 are the circle group T and SU (2).(The group SO(3) has torsion in π 1 and therefore is not covered by Proposition 3.1.)If n = 1, then Theorem 2.6 applies to all objects of A. That is, any M ∈∈ A has a lifting, and equivalence classes of liftings are in bijection with Ext ) with its usual Z/2-grading, so that we may also denote it by Ext 2 R (M, M ) − .The category A is even, so that the results of Section 2.1 apply as well.That is, there is a canonical lifting of any M ∈∈ A, namely, the direct sum M+ ⊕ M− , where M+ and M− are the unique lifting of the even and odd part of M , respectively.Every object A ∈∈ C has an invariant (M, δ) ∈∈ Aδ with M := K G * (A) and δ ∈ Ext 2 R (M, M ) − ; Theorem 2.17 says that A 1 , A 2 ∈∈ C corresponding to (M 1 , δ 1 ) and (M 2 , δ 2 ) in Aδ are isomorphic if and only if there is a grading preserving R-module isomorphism f : then Theorem 2.6 still applies, among others, to objects of A that are free as Abelian groups.Groups G for which this happens are T 2 , T × SU(2), SU(2) × SU(2), SU(3), Spin (5), and the simply connected compact Lie group with Dynkin diagram of type G 2 .For even higher rank, we know no useful sufficient criterion for an R-module to have a projective resolution of length 2.
We now consider some natural examples of circle actions on C * -algebras.Thus G = T and R = Z[x, x −1 ] from now on.
, where (x − n) means the principal ideal generated by x − n.This is concentrated in degree 0 and has a length-1-projective resolution .The fixed-point algebra is an AF-algebra, and its K 0 -group is isomorphic to the direct limit of the iteration sequence For the most useful gauge actions, the fixed-point algebra is AF, however, so that we cannot expect isomorphisms in KK T to lift to * -isomorphisms.
3.2.On the computation of Ext 2 .We now describe Ext * R (V, W ) for two general R-modules V and W , where R = Z[x, x −1 ].We view an R-module V as an Abelian group with an automorphism x V , namely, the action of the generator x ∈ R.
The ring R has a very short R-bimodule resolution This remains exact when we apply the functor ␣ ⊗ R V for a left R-module V , and this gives a short exact sequence of R-modules for any R-module V .Given another R-module W , the long exact cohomology sequence for this short exact sequence becomes ) by adjoint associativity.Thus we get a long exact sequence Here the maps Hom Z (V, W ) → Hom Z (V, W ) and Ext 1 , using the automorphisms x V and x W of V and W . Hence ). Remark 3.6.We may view this cokernel as the first Hochschild cohomology for R with coefficients in Ext 1 Z (V, W ) with the induced R-bimodule structure.The kernel of this map is the zeroth Hochschild cohomology.The above long exact sequence is equivalent to a spectral sequence [13,14] that for circle actions on unital Kirchberg algebras A with the Rokhlin property, such that A satisfies the Universal Coefficient Theorem and has finitely generated K-theory groups, the action of the generator of R on equivariant K-theory is the identity (this is analogous to the situation of finite group actions with the Rokhlin property, see [29]), and equivariant K-theory together with the unit class is a complete invariant.Moreover, K 0 (A) ∼ = K 1 (A) ∼ = K 0 (A T )⊕K 1 (A T ), and every pair (G 0 , G 1 ) of finitely generated Abelian groups with any unit class in G 0 may be realised as K 0 (A T ), K 1 (A T ) .
If x acts identically on V and W , then Ext 2 R (V, W ) ∼ = Ext 1 Z (V, W ) by (3.5).Therefore, there must be a unique obstruction class in Ext 1 Z K * (A T ), K * +1 (A T ) that comes from a continuous Rokhlin action on a unital Kirchberg algebra.We do not know, however, which obstruction class this is.
Another classification result for Rokhlin actions of finite groups on Kirchberg algebras was proved by Masaki Izumi [15].

3.3.
Nekrashevych's C * -algebras of self-similar groups.Nekrashevych [27] constructs purely infinite simple C * -algebras with a gauge action of T from selfsimilar groups.He proves that the conjugacy class of this gauge action essentially determines the underlying self-similar group and hence is a very fine invariant.This is, however, far from true for the KK T -equivalence class.
We consider only the particular case considered in [27,Theorem 4.8] to use Nekrashevych's K-theory computation.The self-similar group G in question is the iterated monodromy group of a post-critically finite, hyperbolic, rational function f on Ĉ.Let n be the (mapping) degree of this rational function, that is, each noncritical point has precisely n preimages.The function f has at most finitely many attracting cycles; let their lengths be ℓ 1 , . . ., ℓ c , listed with repetitions.Thus f has c attracting cycles.
It is asserted in [27,Theorem 4.8] that the K-theory of the gauge fixed-point algebra of the C * -algebra O G associated to f is Z[1/n] in even degrees and Z k−1 in odd degrees, where k = c i=1 ℓ i .We can be more precise: the proof of [27, Theorem 4.8] also gives the T-equivariant K-theory of O G .
First, the fixed-point algebra is Morita-Rieffel equivalent to the crossed product in this case, so that the K-theory of the gauge fixed-point algebra is isomorphic to the T-equivariant K-theory and carries a Z[x, x −1 ]-module structure.The action of x on this module is given by multiplication by n on the even part, as for the Cuntz algebra by the diagonally embedded copy of Z.We may view H as the space of functions from the union of the attracting cycles of f to Z.The generator x acts like f on these functions, that is, it is a cyclic permutation in each copy of Z ℓi .Thus we get the quotient of the module by the copy of Z generated by Since O T G is the C * -algebra of an amenable groupoid, it belongs to the bootstrap class.Hence so does the Morita-Rieffel equivalent C * -algebra O G ⋊ T, so our classification results apply by Proposition 3.1.Lemma 3.7 shows that, up to circle-equivariant KK-equivalence, the C * -algebra O G is classified completely by the . This module only remembers the degree d of f and the multiset of lengths ℓ i , so it is a rather coarse invariant.
It would be very interesting to refine our invariant to detect the conjugacy class of the gauge action because this determines the action of f on its Julia set up to topological conjugacy by Nekrashevych's main result ([27, Section 4.2]).Unfortunately, we know no useful refinements for our invariant.Both for O G and for Cuntz-Krieger algebras, the fixed-point algebra of the gauge action has a unique trace.For Cuntz-Krieger algebras, the order structure on K T 0 gives a finer invariant (see Section 3.1), but for O G , the group K T 0 (O G ) ∼ = Z[1/d] carries no interesting order structure.

Kasparov theory for C * -algebras over unique path spaces
Let X be a finite T 0 -space.In this section, we consider Kirchberg's ideal-related KK-theory T := KK(X), following [24,26].Let i x C ∈∈ T denote the C * -algebra of complex numbers C equipped with the continuous map Prim(C) → X taking the unique element of Prim(C) to x ∈ X.The bootstrap class B(X) in T is the localising subcategory generated by the collection {i x C | x ∈ X} of one-dimensional C * -algebras over X (see [24,Definition 4.11]).
We apply the homological machinery from [25] to the family of functors represented by the objects i x C, respectively.Let A(U x ) be the distinguished ideal of A corresponding to the minimal open neighbourhood U x of x in X.The adjointness relations in [24, Proposition 3.13] specialise to For x ∈ X, consider the stable homological functor A) and the homological ideal x takes the free rank-one Abelian group in even degree to the object i x C. [25, Theorem 57] implies that F x is the universal I x -exact functor and that I x has enough projective objects.Now we consider the homological ideal I := x∈X I x .[24, Theorem 4.17] gives KK * (X; A, B) = 0 for all I-contractible B if and only if A belongs to B(X).By [25, Proposition 55], the ideal I has enough projective objects.An argument as in [26,Section 4.3] shows that the universal I-exact stable homological functor is XK := KK * (X; R, ␣) : T → Mod KK * (X; R, R) op c , where R := x∈X i x C and Mod KK * (X; R, R) op c denotes the category of countable Z/2-graded right modules over the Z/2-graded ring KK * (X; R, R).
Partially order X by x y if and only if x ∈ {y}, if and only if y ∈ U x .Equation (4.1) implies for all x, y ∈ X.The proof of (4.1) shows that the generator of KK 0 (X; i x C, i y C) = Z for x y is the class i y x of the identity map on C, viewed as a * -homomorphism over X from i x C to i y C. Since i z y • i y x = i z x , the Z/2-graded ring KK * (X; R, R) op is isomorphic to the integral incidence algebra Z[X] of the poset (X, ) in even degree and vanishes in odd degree; here we use the convention that Z[X] is the free Abelian group generated by elements f x y for all pairs (x, y) with x y; the multiplication is defined by f x y f y z = f x z .
We write x → y for x, y ∈ X if x ≻ y and there is no z ∈ X with x ≻ z ≻ y.Then x ≻ y if and only if there is a chain If X is a unique path space, then Z[X] is the integral path algebra of the quiver (X, →), where paths are concatenated such that arrows point to the left.
There is a canonical family of orthogonal idempotent elements e x ∈ Z[X] for x ∈ X with x∈X e x = 1.Viewing Z[X]-bimodules as modules over the ring (corresponding to the idempotent e x ⊗ e y ).
Lemma 4.3.Let X be a unique path space.There is a length-one projective bimodule resolution The second map is determined by the maps Proof.We clearly have a complex of bimodule maps.The underlying Abelian groups of the three modules are free.To verify exactness, we choose the following canonical bases.A basis for Z[X] is given by paths p = (x 1 → • • • → x k ) of non-negative length in X; similarly, a basis for the middle group is given by paths with a marked vertex position l ∈ {1, . . ., k}, and a basis for the left group is given by paths p l,l+1 = ( with two marked, consecutive vertices x l and x l+1 .In this picture, the right map simply forgets the position of the marked vertex.Hence it is surjective.The left map takes a doubly marked path p l,l+1 to the linear combination p l − p l+1 of singly marked paths.Since this map does not change the underlying path p, it suffices to check injectivity of the left map on elements of the form In fact, it is easy to see that the cohomological dimension of KK * (X; R, R) op is equal to 2 unless the space X is discrete (in which case it is 1).
Proof.Tensoring the above short exact bimodule sequence over Z[X] with a left Z[X]-module V gives the short exact sequence 0 The long exact cohomology sequence for the functor Hom Z[X] (␣, W ) applied to the above short exact sequence is thus of the form 0 The maps in the exact sequence above are the sum of the maps ), induced by the arrows i : y → x in X.This gives a scheme for computing the groups Ext n Z[X] (V, W ). As in Section 3.2, the above long exact sequence is equivalent to a spectral sequence Combining Theorem 2.17 and Proposition 4.4 with Kirchberg's Classification Theorem in [17], we get the following purely algebraic complete classification of Kirchberg X-algebras in the bootstrap class B(X): Corollary 4.6.Let X be a unique path space.Then the functor XKδ induces a bijection between the set of * -isomorphism classes over X of stable Kirchberg X-algebras in the bootstrap class B(X) and the set of isomorphism classes in the is simply the ring of integers, which has global dimension 1.Hence, in this case, the functor XKδ reduces to plain Z/2-graded K-theory.
Example 4.8.Let X = • → • be the two-point Sierpiński space.Stable Kirchberg X-algebras in B(X) are essentially the same as stable extensions of UCT Kirchberg algebras.Rørdam [32] classified these by their six-term exact sequences in K-theory.Moreover, every six-term exact sequence of countable Abelian groups arises as the K-theory sequence of a stable Kirchberg X-algebra in B(X).The equivalence between Rørdam's invariant and ours becomes obvious by the following direct computation: given two objects The group Ext 1 Z ker(ϕ), coker(ψ) is in natural bijection to the set of equivalence classes of exact sequences of the form In fact, our invariant factors through Rørdam's; it remembers isomorphism classes but forgets certain morphisms.
Example 4.9.Let X be totally ordered (for two points, this is Example 4.8).Then filtrated K-theory is a complete invariant for objects in B(X) by the main result of [26].Since totally ordered spaces are unique path spaces, we now have two seemingly different complete invariants for objects in B(X).Both invariants must contain exactly the same information.The authors, however, do not understand the relationship between these two invariants.If, for instance, such that all squares commute and certain exactness conditions hold.

Graph C * -algebras
If the finite T 0 -space X is not a unique path space, then we may still classify those objects A of B(X) for which XK(A) has a projective resolution of length 2. We are going to show that graph C * -algebras with finitely many ideals have this property.Even better, we may compute their obstruction classes in terms of the Pimsner-Voiculescu type sequence that computes their K-theory.

A computation of obstruction classes.
First we prove a general result in the abstract setting of a triangulated category T with a universal I-exact stable homological functor F : T → A; we also impose the parity assumptions of Section 2.1.For certain objects in T that are constructed from a length-2 projective resolution in A, we compute the obstruction class explicitly.Let be an exact chain complex in A + with projective objects M 1 , Q 1 and Q 0 .The adjoint functor F ⊢ on projective objects of A gives objects M1 , Q1 and Q0 of T lifting M 1 , Q 1 and Q 0 , and maps ∂2 ∈ T( M1 , Q1 ) and ∂1 ∈ T( Q1 , Q0 ) lifting ∂ 2 and ∂ 1 .Embed ∂1 into an exact triangle The long exact sequence for F applied to this triangle has the form This has the following projective resolution of length 2: Theorem 5.3.The obstruction class of A is the class of the 2-step extension (5.1) in Ext 2 A (M 0 , M 1 ), which we embed as a direct summand into be another exact chain complex in A with even projective objects M ′ 1 , Q ′ 1 and Q ′ 0 , and let A ′ be the cone of the lifting ∂′ 1 of ∂ 1 .Then A ∼ = A ′ if and only if there is a commutative diagram (5.5) in A, where the maps ϕ i for i = 1, 2, 3, 4 are isomorphisms.
Proof.We first compute the obstruction class of A. For this, we compare A to the canonical lifting of F (A).The latter is the direct sum of the canonical lifting of M 0 with M1 [1].To lift M 0 canonically, we first embed ∂2 : M1 → Q1 in an exact triangle Then F (D) ∼ = coker ∂ 2 ∼ = ker ε.The UCT gives T 0 (D, Q0 ) ∼ = A(ker ε, Q 0 ) for parity reasons.Hence there is a unique x ∈ T 0 (D, Q0 ) for which F (x) is the inclusion of ker ε into Q 0 .The cone of x is the canonical lifting of M 0 .Since direct sums of exact triangles remain exact, the canonical lifting of F (A) is the cone of the map (x, 0) : where i 1 is the inclusion of the first summand and p 2 the projection onto the second summand.The octahedral axiom applied to ∂1 and ∂2 gives maps x : D → Q0 , y : D → M1 [1] and commutes and has exact rows and columns.We claim that x = x.Recall that F (u) is surjective, so F (x) is determined by its composite with F (u), which is equal to F ( ∂1 ) = ∂ 1 by the commuting diagram.Hence F (x) = F (x), which gives x = x by the uniqueness of x.The exactness of the third column means that A is the cone of the map (x, v) : D → Q0 ⊕ M1 [1].The canonical lifting of F (A) is the cone of the map (x, 0) : D → Q0 ⊕ M1 [1].Hence the obstruction class of A is the image of (x, v) − (x, 0) = (0, v) under the map from Ext constructed in the proof of Theorem 2.6.

We may describe the element in Ext 1
A F (D), Q 0 [−1] ⊕ M 1 induced by (0, v) because v also appears in the first row: it is represented by the extension The next step is to push forward along the map To get the obstruction class for A in Ext 2 A F (A), F (A)[−1] , we need to splice the extension above with the extension where ι : F (D) → Q 0 denotes the inclusion map F (D) ∼ = ker ε ⊆ Q 0 .Up to the identification stated in the theorem, this yields indeed the class of the 2-step extension (5.1).Now we establish the isomorphism criterion.First assume that there are invertible maps ϕ i as in (5.5).Since the cone of the identity map is the zero object and since cones are additive for direct sums, the cone of ∂1 ⊕ id Q′ 0 is again A, and the cone of ∂′ 1 ⊕ id Q0 is again A ′ .Since the maps ∂1 ⊕ id Q′ 0 and ∂′ 1 ⊕ id Q0 are isomorphic by (5.5), they have isomorphic cones.Thus A ∼ = A ′ .
Conversely, assume that A ∼ = A ′ .Then F (A) ∼ = F (A ′ ), so that we get isomorphisms To simplify notation, we assume without loss of generality that ϕ 1 and ϕ 4 are identity maps.Then the isomorphism A ∼ = A ′ is an equivalence of liftings, so that A and A ′ have the same obstruction class in Ext A (M 0 , M 1 ).Since Q 0 and Q ′ 0 are projective, there are maps ψ : is an isomorphism with (0, ε ′ )•ϕ 3 = (ε, 0).Hence ϕ 3 makes the third square in (5.5) commute.
Let K be the kernel of (ε, 0) : . Hence the equality of the obstruction classes shows that the extensions 1 ։ K that we get from the two rows in (5.5) and the isomorphism ϕ 3 have the same class in Ext 1 A (K, M 1 ).Equality in Ext 1 A (K, M 1 ) means that the extensions really are isomorphic in the strongest possible sense, that is, there is an isomorphism ϕ that induces an isomorphism of extensions.This means that it makes the remaining two squares in (5.5) commute.Thus A ∼ = A ′ implies that there are isomorphisms ϕ i making (5.5)commute.
Remark 5.6.The same argument works if the objects in (5.1) all belong to A − .If the objects in (5.1) belong to A, then we may split (5.1) into its even and odd parts.Thus the obvious adaption of Theorem 5.3 still holds without any parity assumptions on the objects M j and Q j .
Remark 5.7.There are several variants of the criterion (5.5).Since (5.1) and (5.4) are exact, isomorphisms ϕ 2 and ϕ 3 making the middle square in (5.5) commute give ϕ 1 and ϕ 4 making all squares in (5.5) commute.Furthermore, if ϕ i are isomorphisms for i = 1, 4 and for i = 2 or i = 3, then the remaining one is an isomorphism as well by the Five Lemma.
If there are maps ϕ i making (5.5)commute, and such that ϕ 1 and ϕ 4 are invertible, then it already follows that A ∼ = A ′ .This is because ϕ 1 and ϕ 4 induce an isomorphism F (A) ∼ = F (A ′ ), and (5.5) shows that the obstruction classes also agree, no matter whether ϕ 2 or ϕ 3 are invertible.5.2.Crossed products for C * -algebras over topological spaces.In this subsection, we generalise some basic results about crossed products to C * -algebras over topological spaces.Let G be a locally compact group.Let X be a second countable topological space.Definition 5.8.A G-C * -algebra over X is a C * -algebra over X whose underlying C * -algebra is a G-C * -algebra such that all distinguished ideals are G-invariant.
In particular, there is a natural KK(X)equivalence between (A ⋊ α G) ⋊ α Ĝ and A.
Proof.We have only added X-equivariance to the classical statement.This follows immediately from the naturality of the classical version (see [7,Theorem 6 in Appendix C of Chapter 2]).
In the following, we assume for convenience that X is finite and A belongs to the category KK(X) loc defined in [24,Definition 4.8].It should be possible to remove these assumptions by carefully checking the naturality of the homotopies effecting the respective equivalences.Proposition 5.10 (Green Imprimitivity).Let H be a closed subgroup of G and let (A, α) be an H-C * -algebra over X. Assume that X is finite and A ∈∈ KK(X) loc .There is a natural KK(X)-equivalence between A ⋊ α H and Ind G H (A, α) ⋊ Ind α G. Proof.By naturality, the imprimitivity bimodule constructed in [10, Theorem 4.1] induces a KK(X)-element which is a pointwise KK-equivalence.By [24, Proposition 4.9], it is a KK(X)-equivalence.Proposition 5.11 (Connes-Thom Isomorphism).Let (A, α) be an R-C * -algebra over X. Assume that X is finite and Proof.We adopt the approach from [19,Proposition 8.3].Let Ã denote the C * -algebra A over X with the trivial R-action.Then C 0 (R, A) and C 0 (R, Ã) with the diagonal actions are naturally * -isomorphic as R-C * -algebras.By [16,Theorem 5.9] we may fix a KK R -equivalence between C[−1] and C 0 (R), where C 0 (R) carries the translation action.In combination, this gives a natural KK R -equivalence between A[−1] and Ã[−1], and consequently also between A and Ã. Taking crossed products gives a natural KK-equivalence between A⋊ α R and A[−1].As in the previous proof, the naturality of the constructed cycle and [24,Proposition 4.9] show that this is a KK(X)-equivalence.Proposition 5.12 (Pimsner-Voiculescu Triangle).Let (A, α) be a Z-C * -algebra over X. Assume that X is finite and A ∈∈ KK(X) loc .Then there is a natural exact triangle in KK(X) of the form Proof.We abbreviate α = α(1) and let has a completely positive X-equivariant section taking a ∈ A to the affine function (1 − t) • a + t • α(a).We get a natural exact triangle in KK(X) of the form (5.13) The R-C * -algebra T α is naturally * -isomorphic over X to Ind R Z (A, α).By Green's imprimitivity theorem and the Connes-Thom isomorphism, we have natural KK(X)equivalences . Plugging this into (5.13) and rotating as appropriate gives an exact triangle of the desired form.The formula for the map from A to A is a consequence of the naturality of the boundary map in the KK-theoretic six-term sequence applied to the morphisms of extensions together with the elementary fact that the extensions Corollary 5.14 (Dual Pimsner-Voiculescu Triangle).Let (A, α) be a T-C * -algebra over X. Assume that X is finite and A ⋊ α T ∈∈ KK(X) loc .Then there is a natural exact triangle in KK(X) of the form Proof.This follows from the Pimsner-Voiculescu Triangle and Takai Duality.

Application to graph algebras.
Let A = C * (E) be the C * -algebra of a countable graph E. We assume that A has only finitely many ideals or, equivalently, that its primitive ideal space is finite; this is necessary for our machinery to work.We set X = Prim(A).The gauge action γ : T A turns A into a T-C * -algebra over X. Corollary 5.14 provides the following natural exact triangle in KK(X): The C * -algebra A ⋊ γ T is AF.Hence the odd part of XK(A ⋊ γ T) vanishes.Applying the functor XK to the dual Pimsner-Voiculescu triangle, we get the following dual Pimsner-Voiculescu exact sequence: The module XK(A ⋊ γ T) is usually not projective, so we cannot directly apply Theorem 5. Proof.The C * -algebra Q0 is an AF-algebra because the graph E × 1 N has no cycle.We claim that all distinguished subquotients of Q0 have free K 0 -groups and vanishing K 1 -groups.Since ideals and quotients of AF-algebras are again AF and since AF-algebras have vanishing K 1 -groups, it suffices to show that the K 0 -group of every distinguished quotient of C * (E× 1 N ) is free.By [2, Corollary 3.5], the quotient of C * (E × 1 N ) by the ideal J HU ×1N,BU ×1N * is isomorphic to the C * -algebra of the graph (E × 1 N )/(H × 1 N ) \ β(B × 1 N * ); its K-theory is free by [2, Lemma 6.2] and continuity of K-theory.Now it follows from [4, Lemma 4.10] that the module XK( Q0 ) is projective.
By an axiom of triangulated categories, there is a morphism f : C s → A[−1] such that the diagram (5.17) commutes.As in [30,Lemma 3.3], it follows that the morphism f * : K * C s (Y ) → K * A(Y ) induced by f is bijective for every closed subset Y ⊆ X.Hence, by [24,Proposition 4.15] and the Five Lemma, it is a KK(X)-equivalence.
Proof.This module is concentrated in even degree and isomorphic to XK 1 C * (E) .Since C * (E) has vanishing exponential maps and all its subquotients have free K 1 -groups, it follows as in [4,Lemma 4.10] that this module is projective.
Since the map f in (5.17) is a KK(X)-equivalence, we get an exact triangle Roughly speaking, stable, purely infinite graph C * -algebras with finitely many ideals are strongly classified by their dual Pimsner-Voiculescu sequence in XK (up to the correct notion of equivalence).
Corollary 5.20.Let A 1 and A 2 be unital, purely infinite graph C * -algebras such that Prim(A 1 ) ∼ = Prim(A 2 ) ∼ = X is finite.Then any isomorphism XKδ(A 1 ) ∼ = XKδ(A 2 ) taking the unit class in K 0 (A 1 ) to the unit class in K 0 (A 2 ) lifts to a * -isomorphism between A 1 and A 2 .
Proof.This follows from [11,Theorem 3.3] and our strong classification theorem up to stable isomorphism.Here we use that, when A has real rank zero, the group K 0 (A) can be naturally recovered from the module XK 0 (A) as a certain cokernel, see [1,Lemma 8.3].
Example 5.21.As in Example 4.8, let X = (2 → 1) be the two-point Sierpiński space.Let A 1 and A 2 be purely infinite tight graph C * -algebras over X such that XK(A 1 ) ∼ = XK(A 2 ).Since the distinguished quotients A 1 ({1}) and A 2 ({1}) have free K 1 -groups it follows using the Smith normal form that A 1 and A 2 have isomorphic six-term sequences.Hence they are stably isomorphic and, in particular, XKδ(A 1 ) ∼ = XKδ(A 2 ).Hence the obstruction class is redundant in this particular case.
We do not expect this phenomenon for more general spaces X.For instance, if X = • → • → • and M is the Z/2-graded Z[X]-module with even part Z ։ Z/25 ։ Z/5 and odd part 0 → 0 → Z, then there are two elements δ 1 and δ 2 in Ext 2 (M, M [−1]) for which (M, δ i ) in Mod Z[X] Z/2 c δ are not isomorphic; both are represented by projective resolutions as in (5.1).However, we cannot prove that the objects (M, δ i ) arise as invariants of tight purely infinite graph C * -algebras over X.We do not know the range of our invariant XKδ on purely infinite graph C * -algebras.
is a direct sum of matrix algebras; similarly, A ⋊ T ∈∈ C .Conversely, assume that A ⋊ T belongs to the usual bootstrap class C ⊆ KK.The assumptions on G imply that H 2 (G, T) = 0. Hence [23, Proposition 3.3] says that any G-C * -algebra A belongs to the localising subcategory of T generated by A ⋊ T equipped with the trivial G-action.Taking the trivial G-action is a triangulated functor

Example 3 . 2 .
Consider the Cuntz algebra O n with its usual gauge action, defined by multiplying each generator by z ∈ T. Then O n ⋊ T is Morita-Rieffel equivalent to the fixed-point algebra O T n .This is the UHF-algebra of type n ∞ .It belongs to the bootstrap class, so that O n ∈ C , and it has K-theory Z[1/n].The generator of the representation ring x acts on this by multiplication by n.Thus

3 . 1 .
Cuntz-Krieger algebras.Now consider the Cuntz-Krieger algebra O A with its usual gauge action; it is defined by an n × n-matrix A with entries in {0, 1} or more generally in the non-negative integers, such that no row or column vanishes identically.The crossed product O A ⋊ T is Morita-Rieffel equivalent to the fixedpoint algebra O T A by [28, Theorem 3.2.2 and Lemma 4.1.1]

[ 6 ,
of the generator x is induced by multiplication with A t (see[8,  Proof of Proposition 3.1]).In particular, given twoCuntz-Krieger algebras O A and O B , for degree reasons we have Ext 2 Z[x,x −1 ] K T Proof.The equivalence of the first two statements follows from the argument above.For the equivalence of the second and third statement, see [20, Theorem 7.5.7].Example 2.13] gives two irreducible non-negative 4 × 4-matrices A and B that are shift equivalent over the integers but not over the non-negative integers.Then the gauge actions on the purely infinite simple Cuntz-Krieger algebras O A and O B are KK T -equivalent by the previous theorem; but the ordered Z[x, x −1 ]modules K 0 (O T A ) and K 0 (O T B ) are not isomorphic by [20, Theorem 7.5.8].Hence the shift automorphisms on the gauge fixed-point algebras O T A and O T B cannot be stably conjugate.By Takai duality, the gauge actions on O A and O B cannot be stably conjugate.We cannot expect a Kirchberg-Phillips type classification result for circle actions unless the fixed-point algebra is also purely infinite and simple.

Proposition 4 . 4 .
this sum can only vanish if n 1 = 0, hence n 2 = 0, and so on.Finally, we show exactness in the middle.The kernel of the right map is generated by elements of the form k l=1 n l p l with k l=1 n l = 0. Rewriting such an element as k−1 l=1 l j=1 n j (p l − p l+1 ) using − k−1 l=1 n l = n k shows that it belongs to the image of the left map.If X is a unique path space, then KK * (X; R, R) op has cohomological dimension at most 2.

3 .Lemma 5 . 16 .
For this purpose, we replace A ⋊ γ T by a suitable C * -subalgebra.This construction is based on the ingredients of the computation of the K-theory of graph C * -algebras in [30, Section 3] and [2, Section 6]; we shall use the notation and a number of results proved in these articles.We may identify C * (E) ⋊ γ T with the C * -algebra of the so-called skew-product graph E × 1 Z.This becomes an isomorphism of C * -algebras over X via the canon-ical definitions C * (E) ⋊ γ T (U ) := C * (E)(U ) ⋊ γ T and C * (E × 1 Z)(U ) := J HU ×Z,BU ×Z where (H U , B U ) is the admissible pair such that C * (E)(U ) = J HU ,BU .We let N denote the set {n ∈ Z | n ≤ 0} and N * = N \ {0}.We let Q0 = Q1 be the C * -subalgebra of C * (E × 1 Z) associated to the subgraph E × 1 N , the restriction of the graph E × 1 Z to the subset of vertices E × N .This is a C * -algebra over X via C * (E × 1 N )(U ) = J HU ×N,BU ×N * , and the inclusion map C * (E × 1 N ) ֒→ C * (E × 1 Z) is a * -homomorphism over X.The module XK( Q0 ) is projective and concentrated in even degree.
[23]all I-contractible B if and only if A belongs to C , the localising subcategory of KK G generated by C; this is the correct analogue of the bootstrap class in this case.Liftings for objects of A are required to belong to C .The following result is implicit in[23].Let G be a connected compact Lie group such that π 1 (G) is torsion-free.Let T be a maximal torus in G.A G-C * -algebra A belongs to the equivariant bootstrap class C if and only if A ⋊ G belongs to the usual bootstrap class in KK, if and only if A ⋊ T belongs to the usual bootstrap class in KK.
A[−1] → Q1 → Q0 → A.Lemmas 5.16 and 5.18show that Theorem 5.3 applies.The obstruction class is the image of the top row in (5.17) under the functor XK.The vertical maps in this diagram show that XK applied to the bottom row also represents the same class in Ext 2 .The bottom row is exactly the dual Pimsner-Voiculescu sequence (5.15), as asserted.Combining these computations with Kirchberg's Classification Theorem gives the following theorem: Theorem 5.19.Let A 1 and A 2 be purely infinite graph C * -algebras such thatPrim(A 1 ) ∼ = Prim(A 2 ) ∼ = X is finite.Then any isomorphism XKδ(A 1 ) ∼ = XKδ(A 2 ) lifts to a stable isomorphism between A 1 and A 2 .The obstruction classes δ(A i ) in Ext 2 XK(A i ), XK(A i )[−1]are determined by the dual Pimsner-Voiculescu sequences (5.15) for the gauge actions γ : T A i .Proof.By [18, Corollary 9.4], a purely infinite, separable, nuclear C * -algebra with real rank zero absorbs the infinite Cuntz algebra O ∞ tensorially.Hence Kirchberg's Classification Theorem applies.It gives the result together with Theorem 5.3.