On the classification of nuclear C^{∗}-algebras

Marius Dadarlat and Søren Eilers

May 1999

Abstract

We employ results from KK-theory, along with quasidiagonality techniques, to obtain wide-ranging classification results for nuclear C^{∗}-algebras. Using a new realization of the Cuntz picture of the Kasparov groups we show that two morphisms inducing equal KK-elements are approximately stably unitarily equivalent. Using K-theory with coefficients to associate a partial KK-element to an approximate morphism, our result is generalized to cover such maps. Conversely, we study the problem of lifting a (positive) partial KK-element to an approximate morphism.

These results are employed to obtain classification results for certain classes of quasidiagonal C^{∗}-algebras introduced by H. Lin, and to reprove the classification of purely infinite nuclear C^{∗}-algebras of Kirchberg and Phillips. It is our hope that this work can be the starting point of a unified approach to classification of nuclear C^{∗}-algebras.

Acknowledgments The first author was partially supported by the National Science Foundation,

# 9622434-DMS. The second author was partially supported by the Carlsberg Foundation.

Subject Classification Primary 46L35, secondary 19K14, 19K35, 46L80.

Contents

1 Introduction 3 1.1 Existence and uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Absorbing and quasidiagonal representations 8 2.1 Two properties of representations . . . . . . . . . . . . . . . . . . . . . 8 2.2 Fundamental examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Absorbing representations (after Kasparov) . . . . . . . . . . . . . . . . 12 2.4 Nuclearly absorbing representations . . . . . . . . . . . . . . . . . . . . 16

3 K-theoretical preliminaries 20 3.1 KK-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2 K-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.3 Partial maps on \underline{K}(-) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4 Uniqueness 30 4.1 Stable uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.2 Some classes of C^{∗}-algebras . . . . . . . . . . . . . . . . . . . . . . . . 34 4.3 Stable uniqueness with bounds . . . . . . . . . . . . . . . . . . . . . . . 36

5 Existence 41 5.1 Realizing group homomorphisms . . . . . . . . . . . . . . . . . . . . . . 42

6 Classification 46 6.1 Approximate unitary equivalence . . . . . . . . . . . . . . . . . . . . . 47 6.2 Classification results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 6.3 Purely infinite C^{∗}-algebras . . . . . . . . . . . . . . . . . . . . . . . . . 59

1 Introduction

Two of the most influential works on C^{∗}-algebras from the mid-seventies — Brown, Douglas and Fillmore’s [3] and Elliott’s [22] — both contain uniqueness and existence results in the now standard sense which we shall outline below. These papers served as keystones for two separate theories —KK-theory and the classification program — which for many years parted ways with only moderate interaction. But with this common origin in mind, it is not surprising that recent years have seen a fruitful interaction which has been one of the main engines behind rapid progress in the classification program.

In the present paper we take this interaction even further. Combining a concept of quasidiagonality for representations and a new characterization of equivalence in the Cuntz picture of KK-theory, we achieve a general uniqueness result. And from the quasidiagonality notion we derive a corresponding general existence result by comparing well-known realizations of the KK-groups. These results are then employed to obtain new classification results for certain classes of quasidiagonal C^{∗}-algebras introduced by H. Lin. An important novel feature of these classes is that they are defined by a certain local approximation property, rather than by an inductive limit construction.

We emphasize that our fundamental uniqueness result does not depend on the universal coefficient theorem, nor on nuclearity of the C^{∗}-algebras involved. On the other hand, when we refine the uniqueness result for use in classification, we need to gradually add such conditions on the C^{∗}-algebras. It is to be expected that requirements of this type are necessary, but we have found that holding them back as long as possible leads to more conceptual proofs while clarifying the role of the UCT and nuclearity in classification. Further, since uniqueness results answer fundamental questions about KK- and K-theory, they are of interest in themselves. We have pursued this theme in [14], and defer to this paper the proof of the new characterization of KK-theory which lies behind our uniqueness results.

Our existence and uniqueness results are in the spirit of the classical Ext-theory from [3]. The main complication overcome in the paper is to control the stabilization which is necessary when one works with finite C^{∗}-algebras. In the infinite case, where programs of this type have already been successfully carried out, stabilization is unnecessary. Yet, our methods are sufficiently versatile to allow us to reprove, from a handful of basic results, the classification of purely infinite nuclear C^{∗}-algebras of Kirchberg and Phillips.

Indeed, it is our hope that this can be the starting point of a unified approach to classification of nuclear C^{∗}-algebras.

1.1 Existence and uniqueness

Existence and uniqueness theorems can be found in most, if not all, classification papers to combine with refinements of Elliott’s intertwining argument to yield classification theorems.

In this framework, a uniqueness result allows one to conclude that if the Ktheoretical elements induced by two ∗-homomorphisms φ, ψ : A ⟶ B coincide, with A, B C^{∗}-algebras satisfying extra conditions, then φ and ψ are equivalent in a sense which generalizes unitary equivalence. In [22], one concludes from φ_{∗} = ψ_{∗} on K_{0}(A) that φ and ψ are approximately unitarily equivalent when A and B are both AF (cf. [6]). In [3] one shows that if [φ]_{KK} = [ψ]_{KK}, then φ and ψ are unitarily equivalent whenever A = C(X) and B = L(H)/K(H).

The role of the equally important existence results is to provide means of realizing given K-theoretical data by a sufficiently multiplicative completely positive contractive map. In [22], one realizes any positive element of Hom(K_{0}(A), K_{0}(B)) by a ∗- homomorphisms when A and B are AF. In [3], any element of KK(C(X), L(H)/K(H)) is realized by a ∗-monomorphism.

One of the main obstacles in achieving general existence and uniqueness results has been the fact that, as soon as one ventures beyond these classical examples, one only can expect to achieve stable versions of such theorems in a sense that involves adding or subtracting maps of the form μ : A ⟶ M_{n}(B). The challenge has been to control the complexity of the stabilization, in order to be able to incorporate them into classification results.

This has been quite successfully carried out, using maps μ with finite-dimensional images, in important classes of quasidiagonal C^{∗}-algebras. But in most existence and uniqueness results so far the source A has been required to be a member of a small class of C^{∗}-algebras forming building blocks for the class of C^{∗}-algebras one has tried to classify, leading to restrictions on the ensuing classification results.

We offer, in the present paper, existence and uniqueness results valid for sources way beyond even the full class of nuclear quasidiagonal C^{∗}-algebras. More precisely, we consider a unital separable source A and a unital target B, and require that this pair allows an nuclearly absorbing and quasidiagonal unital representation γ : A ⟶ M(K(H) ⊗ B) as defined in Section 2 below. Such a map exists automatically when A is quasidiagonal, or by Theorem 2.18 whenever A can be embedded in B such that each a ∈ A, a ≠ 0, generates a dense ideal of B. In this case, our uniqueness result Theorem 4.5 states that if φ, ψ : A ⟶ B are two nuclear ∗-homomorphisms inducing the same KK_{nuc}-class, we may conclude that φ is stably approximately unitarily equivalent to ψ in a sense involving adding “finite pieces” of γ.

As mentioned above, our fundamental uniqueness result does not depend on the universal coefficient theorem of [47], nor on nuclearity of the C^{∗}-algebras involved. For a nuclear source algebra A which satisfies the UCT, our result is predated by one of a similar nature, valid when A or the target algebra is simple, which appears in [34].

In fact, we shall require uniqueness results which hold also for sufficiently multiplicative completely positive contractive maps, and this entails the problem of associating to such maps a kind of partial KK-elements to substitute for the globally defined group homomorphisms one gets from ∗-homomorphisms. This is done using the universal multicoefficient theorem of [19] in a fashion explained by the first named author in his talk at the Workshop on the classification of amenable C^{∗}-algebras at the Fields Institute in December of 1994. As soon as we have made sense out of this concept, a uniqueness result for completely positive contractive maps, Theorem 4.15, can be derived from those for ∗-homomorphisms using a procedure originating with [38] in the torsion free case. This method also requires keeping a close eye on the K-theory for products of C^{∗}-algebras.

In our existence result Theorem 5.5 we manage to realize — partially, in a sense corresponding to the one used in the uniqueness result — a given element from KK_{nuc}(A, B) as a difference of nuclear completely positive contractive maps from A to M_{N}(B). Again, all we require is that there exists a strictly nuclear nuclearly absorbing absorbing quasidiagonal representation γ : A ⟶ M(K(H) ⊗ B). Furthermore, one of the maps in the difference can be chosen as a “finite piece” of γ.

In the building block approach, establishing existence has typically been somewhat easier than achieving uniqueness results. This may still be the case in our setting, but at the current stage it is in fact the existence which is causing problems. Indeed, the existence result offered in our paper has shortcomings in the finite case which are responsible for a number of unwanted, and hopefully redundant, restrictions in the resulting classification results. For instance, we do not have sufficient technology to prove in full generality that a positive KK-element can be realized by a single map, as one would expect to be the case.

As a main application, we apply our existence and uniqueness results to the class of TAF C^{∗}-algebras introduced and studied by H. Lin. Using a factorization property of these C^{∗}-algebras, we are able to prove in Theorem 6.12 that, up to an isomorphism, there is only one unital, separable, nuclear and simple TAF C^{∗}-algebra satisfying the UCT with K_{0}(A) = ℚ and K_{1}(A) = G, where G is a countable fixed arbitrary group, thus proving that every such C^{∗}-algebra falls in the well studied class of AD algebras of real rank zero.

We believe, however, that our existence and uniqueness results will be applicable to a wide range of classification problems, extending well beyond the TAF case. In fact, it might be that at least the uniqueness result will be sufficiently versatile to serve as a unifying element for many future efforts to classify nuclear C^{∗}-algebras. To substantiate this claim, we apply our methods to the case of purely infinite C^{∗}-algebras, reproving in Theorem 6.21, rather easily, the classification theorem of purely infinite simple unital nuclear C^{∗}-algebras (see [30] and [39]) from a handful of fundamental results about such algebras.

We reported on the present paper at the 1998 GPOTS. At the same conference, H. Lin reported results which – although the methods differ – overlap with our classification results in the TAF C^{∗}-algebra case. More details are given in the notes of the present paper. The exposition of the paper, and many intermediate results, have been improved since the first final version as regards the role of nuclearity as described above. However, the end results are the same.

1.2 Methods

Apart from KK-theory, the main technical tools are approximate morphisms and partial KK-elements, defined using K-theory with coefficients.

We arrive at uniqueness and existence results from the existence of an absorbing and quasidiagonal representation. To achieve such representations, one may use the Kasparov Weyl-von Neumann-Voiculescu theorem along with the concept of quasidiagonality. One can also employ a related result by H. Lin [34] stating that a certain representation associated to a unital inclusion ι : A ⟶ B is absorbing when A is nuclear and B is simple. A generalization of this result is offered in Theorem 2.18.

To refine uniqueness results we depend on a number of basic properties about the K-theory of products of C^{∗}-algebras. Furthermore, to apply our results to classify TAF C^{∗}-algebras, we use several results about structural properties of such C^{∗}-algebras by Lin. We prove classification for purely infinite algebras by appealing to the embedding theorem for exact C^{∗}-algebras of [31], as well as structural results by Kirchberg, Phillips and Rørdam.

2 Absorbing and quasidiagonal representations

In the paper, a representation refers to a ∗-homomorphism from a general C^{∗}-algebra to a C^{∗}-algebra of the form L(E), where E is a Hilbert C^{∗}-algebra module. The reader is referred to [29] for an introduction to these objects. Throughout the paper, all Hilbert C^{∗}-algebra modules are assumed to be countable over a σ-unital C^{∗}-algebra B. As usual, the canonical such module is denoted by H_{B}.

In this section we are going to define two properties for representations of a C^{∗}- algebra which are refinements of known properties. Of these notions, the definition of nuclearly absorbing representations is going to ensure that a given representation, in a sense to be made precise, contains the full representation theory of the C^{∗}-algebra, whereas the quasidiagonality property ensures that the representation can be approximated by finite-dimensional compressions. These two properties combine to allow uniqueness and existence results in the following.

2.1 Two properties of representations

Definition 2.1 Fix a σ-unital C^{∗}-algebra B. When γ : A ⟶ L(E) and γ′ : A ⟶ L(E′) are two representations, with E and E′ Hilbert B-modules, we say that γ and γ′ are approximately unitarily equivalent and write γ ∼ γ′, if there exists a sequence of unitaries u_{n} ∈ L(E′, E) such that

(i) lim_{n→∞} ∥γ(a) − u_{n}γ′(a)u_{n}^{∗}∥ = 0,
(ii) γ(a) − u_{n}γ′(a)u_{n}^{∗} ∈ K(E)

for any a ∈ A.

We say that a representation γ is absorbing if γ ∼ γ ⊕ σ for any σ. Here follows a generalization of that notion:

Definition 2.2 ([51, Definition 1.1.d]) A completely positive contraction π : A ⟶ L(E) is said to be strictly nuclear if there exist integers (n_{λ}) and generalized sequences ψ_{λ} : A → M_{n_{λ}}, φ_{λ} : M_{n_{λ}} → L(E) of completely positive contractions such that

lim_{λ⟶∞} φ_{λ}ψ_{λ}(a) = π(a)

in the strict topology, for all a ∈ A.

Remark 2.3 If either A or B is nuclear, then any completely positive contraction π : A ⟶ L(E) is strictly nuclear by [51, 1.7]. Any scalar representation θ : A → L(H) ⊂ L(H_{B}) is strictly nuclear, since it factors through a module over the nuclear C^{∗}-algebra ℂ.

Definition 2.4 A nonunital representation π : A → L(E) is called nuclearly absorbing if π ⊕ σ ∼ π for any strictly nuclear representation σ : A → L(F). A unital representation π : A → L(E) is called nuclearly absorbing if π ⊕ σ ∼ π for any unital strictly nuclear representation σ : A → L(F).

We emphasize that we have two different versions of nuclearly absorbing representations, one for unital representations and a different one for nonunital representations. We are going to explain how they interrelate in Lemma 2.17 below.

Definition 2.5 A representation γ : A ⟶ M(K(H) ⊗ B) is quasidiagonal if there exists an approximate unit of projections (e_{n}) for K(H) ⊗ B with the property that

[e_{n}, γ(a)] ⟶ 0, a ∈ A.

It is clear that if γ is quasidiagonal, one may assume that (e_{n}) has the property that e_{n} ∈ M_{r_{n}}(B) for some sequence of integers (r_{n}). When this is the case, we define a sequence of completely positive maps γ_{n} : A ⟶ M_{r_{n}}(B) by γ_{n}(a) = e_{n}γ(a)e_{n}. We call (γ_{n})_{n∈ℕ} a quasidiagonalization of γ by (e_{n})_{n∈ℕ}. Note that the sequence (γ_{n})_{n∈ℕ} is asymptotically multiplicative, and that it consists of nuclear maps if γ is strictly nuclear, according to Definition 2.4.

The concept of relative quasidiagonality that we use in this paper was studied in [48] and [50].

Remark 2.6 (i) One notes that a nuclearly absorbing representation need not be strictly nuclear.

(ii) If a representation is both strictly nuclear and quasidiagonal, then it follows from [51] that any element of a quasidiagonalization is a nuclear map. Indeed, π is strictly nuclear precisely when for any v ∈ K(E), the map A → K(E), defined by a ↦ v^{∗}π(a)v is nuclear.

(iii) If γ is quasidiagonal and γ ∼ γ′, then γ′ is quasidiagonal. Thus if there exists a strictly nuclear nuclearly absorbing and quasidiagonal representation γ : A ⟶ M(K(H) ⊗ B), then any strictly nuclear nuclearly absorbing representation will be quasidiagonal.

2.2 Fundamental examples

Definition 2.7 Let A be a separable C^{∗}-algebra. An admissible scalar representation θ : A ⟶ M(K(H) ⊗ B) is a ∗-homomorphism which factors as

\xymatrix{A \ar[r]^{θ′} & L(H) \ar[r]^{−⊗1} & L(H) ⊗ M(B) \ar@{^{(}->}[r] & M(K(H) ⊗ B)}

where θ′ is faithful and of infinite multiplicity, i.e. of the form ∞·γ for some representation γ.

Definition 2.8 A unital ∗-homomorphism ι : A ↪ B is called a full embedding if span Bι(a)B is dense in B for all nonzero a ∈ A.

The following definition gives a class of examples of full embeddings.

Definition 2.9 We say that a unital ∗-homomorphism ι : A ⟶ B is a unital simple embedding if it can be factored

\xymatrix{A \ar@{^{(}->}[r]^{ι′} & C \ar@{^{(}->}[r]^{ι′′} & B}

where ι′, ι′′ are injective and C is simple.

This implies that C, ι′ and ι′′ are unital. Note that the composition (on either side) of a simple embedding with a unital injective ∗-homomorphism is a simple embedding. In this setting, we define a representation d_{ι} : A ⟶ M(K(H) ⊗ B) by

d_{ι}(a) = 1 ⊗ ι(a).

With all the definitions in place we can now state the main results of the ensuing sections as follows. Let A be a unital separable C^{∗}-algebra.

(i) If θ : A ⟶ L(H_{B}) is a unital admissible scalar representation, then θ is nuclearly absorbing (Proposition 2.18).

(ii) If ι : A ↪ B is a unital full embedding, then d_{ι} is nuclearly absorbing (Theorem 2.22).

Here (i) follows from the arguments of [29], as pointed out in [51]. As we need many of the elements involved in proving this, we are going to supply details for the benefit of the reader. Further, (ii) is a generalization of a result in [34], valid when ι : A → B is a unital embedding, A is nuclear, and either A or B is simple.

The results combine with the quasidiagonality condition. Indeed, note that d_{ι} is always quasidiagonal, as it commutes with projections e_{n} = n·1_{B}. Thus by Remark 2.6(iii) all unital strictly nuclear nuclearly absorbing representations γ : A ⟶ M(K(H) ⊗ B) are quasidiagonal, when there is a unital nuclear full embedding of A into B.

If A is a quasidiagonal C^{∗}-algebra, then by definition, it has an admissible scalar representation θ : A → M(K(H) ⊗ B) which is quasidiagonal. Therefore any strictly nuclear nuclearly absorbing representation of A is quasidiagonal. We are going to need the following subclass of the quasidiagonal C^{∗}-algebras:

Definition 2.10 Let A be a separable C^{∗}-algebra and let (π_{n}) be a sequence of finite dimensional representations of A. We say that (π_{n}) is a separating sequence if for every a ∈ A there is n such that π_{n}(a) ≠ 0. If this is the case, we say that A is residually finite dimensional or RFD.

Finally, note as in Remark 2.3 the map θ is always strictly nuclear. The map d_{ι} is strictly nuclear precisely when ι is nuclear, cf. Remark 2.6(ii).

2.3 Absorbing representations (after Kasparov)

Definition 2.11 If π : A → L(E) is a representation and φ : A → K(F) is a completely positive map, we write φ ≺ π if there is a bounded sequence v_{i} ∈ K(F, E) such that

(i) lim_{i→∞} ∥φ(a) − v_{i}^{∗}π(a)v_{i}∥ = 0 for all a ∈ A (ii) lim_{i→∞} ∥v_{i}^{∗}ξ∥ = 0 for all ξ ∈ E.

The condition (ii) above is equivalent to

(ii’) lim_{i→∞} ∥v_{i}^{∗}x∥ = 0 for all x ∈ K(E).

If σ : A → L(F) is a representation, we define σ_{∞} : A → L(F_{∞}) by σ_{∞} = σ⊕σ⊕⋯, where F_{∞} = F ⊕ F ⊕ · · ·.

The proof of the following lemma is routine; we leave it to the reader.

Lemma 2.12 Let π and π′ be representations of A on L(E) and L(E′), and let φ and φ′ be completely positive maps from A to K(F) and K(F′). We have

(i) If φ ≺ π and φ′ ≺ π, then φ ⊕ φ′ ≺ π.

(ii) If φ ≺ π, then φ ≺ π ⊕ π′.

(iii) If φ ≺ π, and π ∼ π′, then φ ≺ π′. (iv) If v ∈ K(E), then vπ(·)v^{∗} ≺ π_{∞}.

We can now state the main result of this section:

Theorem 2.13 Let A be a unital separable C^{∗}-algebra and let B be a σ-unital C^{∗}- algebra. Let π : A → L(E) and σ : A → L(F) be two unital representations where E, F are Hilbert B-modules. The following assertions are equivalent:

(i) π ∼ σ_{∞} ⊕ π.

(ii) For any v ∈ K(F), if φ = v^{∗}σ(·)v, then φ ≺ π.

(iii) There is an increasing approximate unit (e_{n}) of K(F), with e_{1} = 0 and quasicentral in σ(A) + K(F), such that (e_{n+k} − e_{n})^{1/2}σ(·)(e_{n+k} − e_{n})^{1/2} ≺ π for all n, k ≥ 1.

The proof of the theorem, given at the end of the section, is based on the following three results, due essentially to Kasparov [29].

Lemma 2.14 Let A, B be as in Theorem 2.13, and let σ : A → L(F) be a representation. Let (e_{n}) be an increasing approximate unit of K(F), quasicentral in σ(A)+K(F). Then for any compact subset F ⊂ A and any ε > 0 there is a subsequence (e_{n(k)}) of (e_{n}) such that if f_{k} = (e_{n(k+1)} − e_{n(k)})^{1/2}, then

(i) λ(a) = ∑_{k=1}^{∞} f_{k}σ(a)f_{k} is strictly convergent, for all a ∈ A.
(ii) σ(a) − λ(a) ∈ K(F) for all a ∈ A.
(iii) ∥σ(a) − λ(a)∥ < ε for all a ∈ F.

Proof: This is a reformulation of [29, Lemma 10]. ■

Proposition 2.15 Let A, B, π : A → L(E) and σ : A → L(F) be as in Theorem 2.13, satisfying condition (iii) there. Then there is a sequence of isometries v_{i} ∈ L(F, E) such that

(i) v_{i}σ(a) − π(a)v_{i} ∈ K(F, E)
(ii) lim_{i→∞} ∥v_{i}σ(a) − π(a)v_{i}∥ = 0 for all a ∈ A.

Proof: The result is implicit in the proof of [29, Theorem 5] and the remark following it. We outline the proof for the benefit of the reader. Since σ is a ∗-homomorphism, it suffices to find a sequence of isometries v_{i} with

lim_{i→∞} ∥σ(a) − v_{i}^{∗}π(a)v_{i}∥ = 0, and σ(a) − v_{i}^{∗}π(a)v_{i} ∈ K(F), a ∈ A.

Indeed, as in [2], this follows from the identity

(v_{i}σ(a) − π(a)v_{i})^{∗}(v_{i}σ(a) − π(a)v_{i}) =
σ(a^{∗}a) − v_{i}^{∗}π(a^{∗}a)v_{i} + (σ(a^{∗}) − v_{i}^{∗}π(a^{∗})v_{i})σ(a) + σ(a^{∗})(σ(a) − v_{i}^{∗}π(a)v_{i})

Fix ε > 0 and F ⊂ A a compact subset with dense span, such that 1_{A} ∈ F = F^{∗}. We need to find an isometry v ∈ L(F, E) such that

∥σ(a) − v^{∗}π(a)v∥ < h(ε) and σ(a) − v^{∗}π(a)v ∈ K(F) (2.1)

for all a ∈ F. Here h is a universal function with lim_{ε→0} h(ε) = 0.

Let (e_{n}) be as in condition (iii) of Theorem 2.13. Let λ(a) = ∑_{k=1}^{∞} f_{k}σ(a)f_{k} be given by Lemma 2.14. Using condition (iii), we find inductively a sequence z_{k} ∈ K(F, E) such that for all a ∈ F and j, k

∥f_{k}σ(a)f_{k} − z_{k}^{∗}π(a)z_{k}∥ ≤ ε · 2^{−k},
∥z_{k}^{∗}π(a)z_{k}∥ ≤ ε · 2^{−j−k} for j < k,
∥z_{k}^{∗}e_{j}∥ ≤ 2^{−k} for j ≤ k.

Then, as in the proof of [29, Theorem 5], one shows that ∑_{k=1}^{∞} z_{k} converges strictly to an element z ∈ L(F, E) satisfying

∥σ(a) − z^{∗}π(a)z∥ < 3ε and σ(a) − z^{∗}π(a)z ∈ K(F)

for all a ∈ F. Since σ(1) = π(1) = 1 and 1 ∈ F, ∥z^{∗}z − 1∥ < 3ε. To obtain (2.1), one replaces z by the isometry v = z(z^{∗}z)^{−1/2}. ■

Let w_{∞} : F_{∞} → F ⊕ F_{∞} be defined as w_{∞}(ξ_{1}, ξ_{2}, ξ_{3}, …) = ξ_{1} ⊕ (ξ_{2}, ξ_{3}, …).

Lemma 2.16 Let π : A → L(E) and σ : A → L(F) be two representations. Then for any isometry v : F_{∞} → E, the unitary u = (1_{F} ⊕ v)w_{∞}v^{∗} + 1_{E} − vv^{∗} ∈ L(E, F ⊕ E) satisfies

∥σ(a) ⊕ π(a) − uπ(a)u^{∗}∥ ≤ 6∥vσ_{∞}(a) − π(a)v∥ + 4∥vσ_{∞}(a^{∗}) − π(a^{∗})v∥.

Moreover, if vσ_{∞}(a) − π(a)v ∈ K(F_{∞}, E), for a ∈ A, then σ(a) ⊕ π(a) − uπ(a)u^{∗} ∈ K(F ⊕ E), for a ∈ A.

Proof: Although the proof of the lemma is implicit in the proof of [29, Theorem 6], we supply a complete argument. Note that the two summands of u are partial isometries. If p = vv^{∗} and p^{⊥} = 1_{E} − p, then (1_{F} ⊕ v)w_{∞}v^{∗} : pE → F ⊕ pE and p^{⊥} : p^{⊥}E → p^{⊥}E act as unitaries. Abbreviate ω(a) = ∥vσ_{∞}(a) − π(a)v∥.

First assume that π(a) and p commute. We have

uπ(a)u^{∗} = (1_{F} ⊕ v)w_{∞}v^{∗}pπ(a)pvw_{∞}^{∗}(1_{F} ⊕ v^{∗}) ⊕ p^{⊥}π(a)p^{⊥}
≈_{ω} (1_{F} ⊕ v)w_{∞}σ_{∞}(a)w_{∞}^{∗}(1_{F} ⊕ v^{∗}) ⊕ p^{⊥}π(a)p^{⊥}
= σ(a) ⊕ vσ_{∞}(a)v^{∗} ⊕ p^{⊥}π(a)p^{⊥}
≈_{ω} σ(a) ⊕ pπ(a)p ⊕ p^{⊥}π(a)p^{⊥}
= σ(a) ⊕ π(a),

where “≈_{ω}” indicates that the error is dominated in norm by ω(a) and that it is compact if vσ_{∞}(a) − π(a)v is.

For general π(a), we consider instead

π′(a) = pπ(a)p ⊕ p^{⊥}π(a)p^{⊥}.

Note that ω′(a) = ∥vσ_{∞}(a) − π′(a)v∥ ≤ ω(a) and that vσ_{∞}(a) − π′(a)v is compact under the compactness assumption on v. We have that ∥σ_{∞}(a) − v^{∗}π(a)v∥ ≤ ω(a), so that

∥p^{⊥}π(a)p∥ ≤ ∥π(a)p − pπ(a)∥ ≤ ∥π(a)vv^{∗} − vσ_{∞}(a)v^{∗}∥ + ∥vσ_{∞}(a)v^{∗} − vv^{∗}π(a)∥
≤ ω(a) + ω(a^{∗});

it is also clear that p^{⊥}π(a)p is compact if vσ_{∞}(a) − π(a)v is. This completes the proof by reducing it to the first case, up to an error of 4(ω(a) + ω(a^{∗})) which is compact under the compactness assumption on v. ■

Proof of Theorem 2.13: Using three observations from Lemma 2.12 we get (i) ⟹ (ii) via

φ = v^{∗}σ(·)v ≺ σ_{∞} ⊕ π ∼ π.

If φ = v^{∗}σ(·)v ≺ σ_{∞}, then φ ≺ σ_{∞} ⊕ π, hence φ ≺ π by Lemma 2.12(iii). The implication (ii) ⟹ (iii) is obvious; take (e_{n}) to be any countable approximate unit of K(F), quasicentral in σ(A) + K(F) (cf. [2]). To prove (iii) ⟹ (i) we first notice that if σ,π and e_{n} ∈ K(F) satisfy (iii), then so do σ_{∞}, π and

e′_{n} = \underbrace{e_{n} ⊕ ⋯ ⊕ e_{n}}_{n} ⊕0 ⊕ ⋯ ∈ K(F_{∞})

by Lemma 2.12(i). Since (σ_{∞})_{∞} = σ_{∞}, from Proposition 2.15 we find a sequence of isometries v_{i} ∈ L(F_{∞}, E) such that

v_{i}(σ_{∞})_{∞}(a) − π(a)v_{i} ∈ K(F_{∞}, E) and lim_{i→∞} ∥v_{i}(σ_{∞})_{∞}(a) − π(a)v_{i}∥ = 0

for all a ∈ A. Let (u_{i}) be a sequence of unitaries given by Lemma 2.16 applied to π, σ_{∞} and v_{i}. Then

σ_{∞}(a) ⊕ π(a) − u_{i}π(a)u_{i}^{∗} ∈ K(F ⊕ E) and lim_{i→∞} ∥σ_{∞} ⊕ π(a) − u_{i}π(a)u_{i}^{∗}∥ = 0

for all a ∈ A. Thus σ_{∞} ⊕ π ∼ π. ■

2.4 Nuclearly absorbing representations

If π : A → L(E) is a representation, we denote by Ã the C^{∗}-algebra obtained by adjoining an external unit ˜1 to A, and define π̃ : Ã → L(E) by π̃(a+λ˜1) = π(a)+λ1_{E}.

A unital representation γ : A → M(K(H) ⊗ B) cannot absorb a nonunital representation. However, if γ̂ = 0 ⊕ γ : A → M(K(H ⊕ H) ⊗ B), where the zero summand acts on H ⊗ B, then we have the following result.

Lemma 2.17 Let A be a separable C^{∗}-algebra and let B be a σ-unital C^{∗}-algebra. If π is a nonunital nuclearly absorbing representation, then π̃ is a unital nuclearly absorbing representation. Suppose that A is unital. If γ : A → M(K(H) ⊗ B) is a unital nuclearly absorbing representation, then γ̂ is a nonunital nuclearly absorbing representation.

Proof: The first part of the lemma is obvious. For the second part, we need to show that if φ : A → L(E) is a strictly nuclear representation, then φ ⊕ γ̂ ∼ γ̂. If p = φ(1), then E ≅ pE ⊕ p^{⊥}E and φ = φ_{1} ⊕ 0_{p^{⊥}E}, with φ_{1} = pφ(·)p. We have φ_{1} ⊕ γ ∼ γ, since γ is nuclearly absorbing. Since p^{⊥}E ⊕ H_{B} ≅ H_{B} by [29, Theorem 2], we obtain

φ ⊕ γ̂ = φ_{1} ⊕ 0_{p^{⊥}E} ⊕ 0_{H_{B}} ⊕ γ ∼ φ_{1} ⊕ γ ⊕ 0_{H_{B}} ∼ 0_{H_{B}} ⊕ γ = γ̂.
■

Proposition 2.18 Let A be a unital separable C^{∗}-algebra. If θ : A ⟶ L(H_{B}) is a unital admissible scalar representation, then θ is nuclearly absorbing.

Proof: Given a unital strictly nuclear representation σ : A → L(F), we want to show that σ ⊕ θ ∼ θ. Assume first that F = H_{B}. We are going to use the implication (ii)⟹ (i) of Theorem 2.13. If v ∈ K(H_{B}) and φ = v^{∗}σ(·)v, we need to show that φ ≺ θ. The set of all completely positive maps φ : A → K(H_{B}) with φ ≺ θ is closed in the point-norm topology. Thus, without loss of generality, we may assume that v ∈ M_{n}(B) for some n. Therefore φ : A → M_{n}(B). Since φ is a nuclear map, because σ is strictly nuclear, φ ≺ θ by [29, Theorem 4]. Hence σ_{∞} ⊕ θ ∼ θ by Theorem 2.13. This implies σ ⊕ θ ∼ θ.

The case F ≅ H_{B} follows from the previous case, after replacing σ by uσ(·)u^{∗}, where u is a unitary implementing the isomorphism F ≅ H_{B}. Indeed, σ⊕θ ∼ uσ(·)u^{∗}⊕θ ∼ θ. If F ≇ H_{B}, we define σ′ = σ ⊕ θ : A → L(F ⊕ H_{B}). Since F ⊕ H_{B} ≅ H_{B} by [29], σ′_{∞} ⊕ θ ∼ θ by the previous part of the proof. Since σ ⊕ σ′_{∞} ∼ σ′_{∞}, we infer that σ ⊕ θ ∼ θ.

■

Proposition 2.19 Let A be a unital, separable C^{∗}-algebra, and let B be a unital C^{∗}- algebra. Let π : A → L(E) be a unital representation. Suppose that θ : A ⟶ L(H_{B}) is a unital admissible scalar representation. Then the following are equivalent:

(i) π is nuclearly absorbing.

(ii) π ∼ θ ⊕ π

(iii) For any unital completely positive map φ : A ⟶ M_{n}(ℂ) ⊆ L(B^{n}), we have φ ≺ π.

Proof: To prove (ii)⟹ (i), note that θ ⊕ σ ∼ θ by Proposition 2.18 again. Using (ii) twice, we get

π ∼ π ⊕ θ ∼ π ⊕ θ ⊕ σ ∼ π ⊕ σ.

The opposite implication is clear since θ is strictly nuclear, and (ii)⟹(iii) follows again from [29, Theorem 4], since by Lemma 2.12(iii), φ ≺ π ⊕ θ ∼ π implies φ ≺ π. It remains to prove (iii)⟹(ii), and we shall do so using the implication (iii) ⟹(i) from Theorem 2.13 with σ = θ. Embed M_{n}(ℂ) into K(H) under an increasing sequence of projections (f_{n}), and let e_{n} = f_{n} ⊗ 1_{B} ∈ K(H_{B}). Then (e_{n}) is an approximate unit of K(H_{B}) which is quasicentral in L(H_{B}). For any n, i we write

(e_{n+i} − e_{n})^{\frac{1}{2}} θ(·)(e_{n+i} − e_{n})^{\frac{1}{2}} = φ_{n,i}(·) ⊗ 1_{B},

with φ_{n,i} : A ⟶ M_{i}(ℂ) a unital completely positive map. Our assertion gives that φ_{n,i}(·) ⊗ 1_{B} ≺ π, so by Theorem 2.13 we conclude that π ⊕ θ_{∞} ∼ π. Hence also π ⊕ θ ∼ π. ■

The following result can be found in [1]:

Lemma 2.20 Let A be a unital separable C^{∗}-algebra and let φ ∈ P (A) be a pure state. Then there is a sequence (x_{n}) in A^{+} with ∥x_{n}∥ = 1 and

∥x_{n}(a − φ(a))x_{n}∥ → 0, ∀a ∈ A.

A sequence (x_{n}) with these properties is said to excise the state φ.

Lemma 2.21 Let A be a unital separable C^{∗}-algebra and let ι : A ↪ B be a unital full embedding. Then φ ≺ d_{ι} for any unital completely positive map φ : A ⟶ M_{n}(ℂ) ⊆ M_{n}(B) = L(B^{n}).

Proof: To simplify notation we identify A with ι(A). Fix a finite set F ⊆ A and a positive ε. It suffices to find an isometry V ∈ L(B^{n}, B^{m}) with

∥φ(a) − V^{∗}(a ⊗ 1_{m})V ∥ < ε a ∈ F (2.2)

We may assume that in fact n = 1 after replacing φ by \overline{φ} : M_{n}(A) ⟶ B as in [29, Lemmas 5 & 6], for M_{n}(A) ↪ M_{n}(B) is also a full embedding.

First assume that φ is a pure state. Apply Lemma 2.20 to get an excising sequence (x_{n}) for φ, and choose n_{0} such that x = x_{n_{0}} satisfies

∥φ(a)x^{2} − xax∥ < ε a ∈ F. (2.3)

For every η ∈ (0, 1) we define f_{η}, g_{η} : [0, 1] ⟶ [0, 1] by

f_{η}(t) = \{ \begin{matrix}0 & t = 0 \\ 1 & t ≥ η \\ affine & 0 ≤ t ≤ η \end{matrix} g_{η}(t) = \{ \begin{matrix}0 & t ≤ η \\ 1 & t ≥ \frac{η+1}{2} \\ affine & η ≤ t ≤ \frac{η+1}{2} \end{matrix}

Note that for a sufficiently large η_{0}, x = f_{η_{0}}(x_{n_{0}}) also satisfies (2.3). Hence if we let y = g_{η_{0}}(x_{n_{0}}), we get a pair of positive norm-one elements x, y such that xy = yx = y and x satisfies (2.3). Since A ↪ B is a full embedding and y ≠ 0 there are b_{1}, …, b_{m} ∈ B such that

b_{1}^{∗}y^{2}b_{1} + ⋯ + b_{m}^{∗}y^{2}b_{m} = 1_{B}.

This is proved in the same way as [9, 1.10].

Define V ∈ L(B, B^{m}) by arranging yb_{1}, …, yb_{m} into a column matrix. Then V is an isometry because V^{∗}V = 1_{B}, and since y acts as a unit on x, we have (x ⊗ 1_{m})V = V. This implies

∥φ(a)1_{B} − V^{∗}(a ⊗ 1_{m})V ∥ = ∥φ(a)V^{∗}(x^{2} ⊗ 1_{m})V − V^{∗}(x ⊗ 1_{m})(a ⊗ 1_{m})(x ⊗ 1_{m})V ∥
≤ ∥φ(a)(x^{2} ⊗ 1_{m}) − xax ⊗ 1_{m}∥ < ε

for any a ∈ F, using (2.3).

For φ an arbitrary state, we argue as follows. By Krein-Milman we may approximate φ by a convex combination of pure states, say

∥φ(a) − ∑_{i=1}^{k} λ_{i}φ_{i}(a)∥ < ε, a ∈ F

Let V_{i} ∈ L(B, B^{mk}) be given by the first part of the proof, with

V_{i}^{∗}V_{i} = 1_{B} V_{i}V_{i}^{∗} ≤ (0, …, 0, \underbrace{1_{m} ⊗ 1_{B}}_{i}, 0, …, 0)

Then V = ∑_{i=1}^{k} \sqrt{λ_{i}}V_{i} defines an isometry of L(B, B^{mk}), such that

∥φ(a) − V^{∗}(a ⊗ 1_{mk})V ∥ = ∥φ(a) − ∑_{i=1}^{k} λ_{i}V_{i}^{∗}(a ⊗ 1_{m})V_{i}∥
≤ ∥φ(a) − ∑_{i=1}^{k} λ_{i}φ_{i}(a)∥ + ∑_{i=1}^{k} λ_{i}∥φ_{i}(a) − V_{i}^{∗}(a ⊗ 1_{m})V_{i}∥
< 2ε,  a ∈ F
■

Theorem 2.22 Let A be a unital separable C^{∗}-algebra and let ι : A ↪ B be a unital full embedding. Then d_{ι} is nuclearly absorbing. Proof: Combine Lemma 2.21 and Proposition 2.19. ■

3 K-theoretical preliminaries

In this section we collect preliminary results and definitions behind our use of K- and KK-theory.

3.1 KK-theory

In all of the paper, except in the proof of the lemma below, we are going to work exclusively with the Cuntz picture of the Kasparov groups KK(A, B). We are only going to give a brief description of this realization here; more details are given in [14], but for complete information we refer the reader to the original sources [11], [25] and [51], or to [28] for a detailed exposition.

The Cuntz picture is described in terms of pairs of representations (φ, ψ) : A → M(K(H) ⊗ B), where

φ(a) − ψ(a) ∈ K(H) ⊗ B, a ∈ A.

Such a pair will be called a Cuntz pair. They form a set denoted by 𝔼_{h}(A, B). A homotopy of Cuntz pairs consists of a Cuntz pair (Φ, Ψ) : A → M(K(H) ⊗ B[0, 1]), and the quotient of 𝔼_{h}(A, B) by homotopy equivalence becomes the Kasparov group KK(A, B).

One defines groups KK_{nuc}(A, B) by restricting attention to Cuntz pairs (and homotopies) consisting of strictly nuclear representations. Hence KK = KK_{nuc} when A or B is nuclear. And one proves:

Lemma 3.1 Let A be a unital separable C^{∗}-algebra and let B be a unital C^{∗}-algebra. Let an admissible scalar representation θ be given. Then any α ∈ KK_{nuc}(A, B) is represented by a strictly nuclear Cuntz pair of the form (ρ, θ̂).

Proof: We use that KK_{nuc}(A, B) also has a Fredholm realization based on triples (φ_{0}, φ_{1}, u) consisting of ∗-homomorphisms φ_{i} : A ⟶ L(E_{i}) intertwined up to compacts by u ∈ L(E_{0}, E_{1}) acting as a unitary, up to compacts, on the ranges of φ_{i}. In fact, according to [51, 2.6], α is represented by a triple of the form (θ̂, θ̂, x). By using the standard simplification given by Proposition 17.4.3 of [4] we may assume that x is a contraction. Finally we may replace x by the unitary

u = [ \begin{matrix}x & (1−xx^{∗})^{1/2} \\ −(1−x^{∗}x)^{1/2} & x^{∗} \end{matrix} ]

Hence, in KK(A, B) we have α = [θ̂, θ̂, u] = [uθ̂u^{∗}, θ̂, 1], and this translates in the Cuntz picture to the desired identity α = [uθ̂u^{∗}, θ̂]. ■

Note that any ∗-homomorphism ψ : A ⟶ B induces an element [φ] ∈ KK(A, B) via the Cuntz pair (ψ, 0). Similarly any nuclear ∗-homomorphism ψ : A ⟶ B induces an element [φ] ∈ KK_{nuc}(A, B).

3.2 K-theory

It is well known, although perhaps not as well known as it should be, that the natural map K_{∗}(∏ B_{i}) → ∏ K_{∗}(B_{i}) is not an isomorphism in general. In this section we are going to study, for large classes of C^{∗}-algebras B_{i}, injectivity properties of the natural map

\underline{ηηη} : \underline{K} (∏ B_{i}) ⟶ ∏ \underline{K} (B_{i})

defined by collecting the maps η^{i}, η_{n}^{i} induced by the projections π_{i} : ∏ B_{i} ⟶ B_{i}. We also use this information to prove that surprisingly often, Pext(−, K_{∗}(∏ B_{i}/ ∑ B_{i})) will vanish.

Here \underline{K} (−) denotes the sum of all K-theory groups with ℤ/n coefficients, n ≥ 1. When discussing maps from \underline{K} (A) to \underline{K} (B), we are only interested in collections of group homomorphisms which preserve the natural set of coefficient transformations and the Bockstein operations (see [49] and [18]). We denote the group of such maps by Hom_{Λ}(\underline{K} (A) , \underline{K} (B)).

If B is a C^{∗}-algebra we denote by Proj(B) the set of all selfadjoint projections in B. The K-theory class of a projection p is denoted by [p] ∈ K_{0}(B). If B is unital we let U_{n}(B) denote the unitary group of M_{n}(B).

K-theory of products

We are going to define five quantities cco (“cancellation order”), pfo (“perforation order”), elo_{0}, elo_{1} (“element lifting order”) and ipo (“infinite height perturbation order”) in ℕ ∪ {∞} for any unital C^{∗}-algebra B by declaring

cco(B) ≤ ℓ if whenever p, q ∈ B ⊗ K, then [p] = [q] ⟹ p ⊕ ℓ·1_{B} ∼ q ⊕ ℓ·1_{B}

pfo(B) ≤ ℓ if for any x ∈ K_{0}(B) such that nx ≥ 0 for some n > 0, one has x + ℓ[1_{B}] ≥ 0.

elo_{0}(B) ≤ ℓ if the canonical map Proj(M_{ℓ}(B)) ⟶ K_{0}(B) is surjective

elo_{1}(B) ≤ ℓ if the canonical map U_{ℓ}(B) ⟶ K_{1}(B) is surjective

ipo(B) ≤ ℓ if for any x in K_{0}(B) and any n ≠ 0, there is y in K_{0}(B) such that −ℓ[1_{B}] ≤ y ≤ ℓ[1_{B}] and x − y ∈ nK_{0}(B).

and declaring the value to be ∞ when no such ℓ exists. When {B_{i}}_{i∈I} is a family of C^{∗}- algebras, we define cco({B_{i}}) = sup_{i} cco(B_{i}), and so forth. We also write rr({B_{i}}) = 0 when each B_{i} has real rank zero.

The following results – some of which are known, cf. [21] – demonstrate the relevance of these quantities to the map \underline{ηηη}. We denote the unit of B_{i} by 1_{i} and the units of ∏ B_{i} and ∏ B_{i}/ ∑ B_{i} by 1_{Π} and 1_{Π/Σ}, respectively.

Lemma 3.2 Let B_{i} be a sequence of unital C^{∗}-algebras for which cco({B_{i}}) < ∞. Then η^{0} is injective, and the image of η^{0} equals

{(x_{i}) ∈ ∏ K_{0}(B_{i})| ∃M ∈ ℕ∀i ∈ ℕ : −M[1_{i}] ≤ x_{i} ≤ M[1_{i}]} .

Furthermore, if η^{0}(x) = (x_{i}) with each x_{i} ≥ 0, then x ≥ 0. If in addition pfo({B_{i}}) < ∞, then Im η^{0} is a pure subgroup of ∏ K_{0}(B_{i}). If elo_{0}({B_{i}}) < ∞, then η^{0} is surjective.

Proof: Assume that cco({B_{i}}) ≤ ℓ. To prove injectivity, let x = [(p_{i})] − [(q_{i})] ∈ K_{0}(∏ B_{i}) be given by p_{i}, q_{i} ∈ M_{N}(B_{i}) and assume that η^{0}(x) = 0. We have that p_{i} ⊕ ℓ·1_{i} ∼ q_{i} ⊕ ℓ·1_{i}, wherefrom (p_{i}) ⊕ ℓ·1_{Π} ∼ (q_{i}) ⊕ ℓ·1_{Π}, proving x = 0. To prove that the image is contained in the set of bounded sequences, write x ∈ K_{0}(∏ B_{i}) as x = [(p_{i})] − [(q_{i})] with p_{i}, q_{i} ∈ M_{N}(B_{i}). By definition of positivity, −N[1_{Π}] ≤ x ≤ N[1_{Π}] and we get the result by applying π_{i}. For the other inclusion, assume that x_{i} ∈ K_{0}(B_{i}) is given with −M[1_{i}] ≤ x_{i} ≤ M[1_{i}]. We can write x_{i} + M[1_{i}] = [r_{i}] for some projection r_{i} in B_{i} ⊗ K, and since [r_{i}] ≤ 2M[1_{i}] we get that 2M[1_{i}] = [r_{i} ⊕ s_{i}] for some projection s_{i} in B_{i} ⊗K. By cco({B_{i}}) ≤ ℓ we see that (2M +ℓ)·1_{i} ∼ r_{i} ⊕s_{i} ⊕ℓ·1_{i}, hence there is q_{i} ∈ M_{2M+ℓ}(B) with r_{i} ∼ q_{i}. Consequently, x_{i} = [r_{i}] − M[1_{i}] can be represented as a difference of projections [q_{i}] − M[1_{i}] where q_{i} ∈ M_{2M+ℓ}(B_{i}). Defining q = (q_{i}) ∈ M_{2M+ℓ}(∏ B_{i}), [q] − M[1_{Π}] is a preimage of (x_{i}).

If η^{0}(x) = (x_{i}) and every x_{i} is positive, we have 0 ≤ x_{i} ≤ M[1_{i}] for some fixed M. Hence x_{i} = [p_{i}] for some p_{i} which we may assume lies in M_{M+ℓ}(B_{i}) as above. Consequently p = (p_{i}) defines an element of K_{0}(∏ B_{i}), and x = [p] by injectivity of η^{0}. To establish purity when pfo({B_{i}}) ≤ ℓ′, assume that x = ny in ∏ K_{0}(B_{i}), where x ∈ Im η^{0} so that for some M, Mn[1_{i}] ± x_{i} ≥ 0. We conclude that (M + ℓ′)[1_{i}] ± y_{i} ≥ 0, whence y ∈ Im η^{0}. Proving surjectivity of η^{0} when elo_{0}({B_{i}}) is finite is straightforward. ■

Lemma 3.3 Let B_{i} is a sequence of C^{∗}-algebras. If rr({B_{i}}) = 0, then η^{1} is injective. If elo_{1}({B_{i}}) < ∞, then η^{1} is surjective.

Proof: To prove injectivity, we assume that η^{1}(x) = 0 with x = [(u_{i})] and u_{i} ∈ M_{N}(B_{i}). By [33], u_{i} is homotopic to n·1_{i} within U_{n}(B_{i}), and since M_{n}(B_{i}) has finite exponential length because rr(B_{i}) = 0 (see [32]), we can combine these paths to one from (u_{n}) to the unit of U_{n}(∏ B_{i}). Proving surjectivity of η^{1} when elo_{1}({B_{i}}) is finite is straightforward. ■

Lemma 3.4 Let B_{i} be a sequence of unital C^{∗}-algebras and abbreviate Π = ∏ B_{i}, Π/Σ = ∏ B_{i}/ ∑ B_{i}. Then

(i) cco(Π), cco(Π/Σ) ≤ cco({B_{i}}).
(ii) pfo(Π), pfo(Π/Σ) ≤ pfo({B_{i}}) if cco({B_{i}}) < ∞.
(iii) elo_{0}(Π), elo_{0}(Π/Σ) ≤ elo_{0}({B_{i}}) if cco({B_{i}}) < ∞.

(iv) elo_{1}(Π), elo_{1}(Π/Σ) ≤ elo_{1}({B_{i}}) if rr({B_{i}}) = 0.
(v) ipo(Π), ipo(Π/Σ) ≤ ipo({B_{i}}) if cco({B_{i}}), pfo({B_{i}}) < ∞.

Proof: These claims all follow in a straightforward fashion from the properties of η^{∗} established in Lemma 3.2 and 3.3. We only prove (v), which is the most involved result. Let x ∈ K_{0}(∏ B_{i}) and n be given and assume that ipo({B_{i}}) ≤ ℓ. With η^{0}(x) = (x_{i}), we can find y_{i} ∈ K_{0}(B_{i}) with −ℓ[1_{i}] ≤ y ≤ ℓ[1_{i}] and x_{i} − y_{i} ∈ nK_{0}(B_{i}). We know from Lemma 3.2 that (y_{i}) = η^{0}(y) for some y ∈ K_{0}(∏ B_{i}) with −ℓ[1_{Π}] ≤ y ≤ ℓ[1_{Π}]. Since η^{0}(x − y) ∈ n ∏ K_{0}(B_{i}) by construction, we further conclude by purity that η^{0}(x − y) ∈ n Im η^{0} and, since η^{0} is injective, that x − y ∈ nK_{0}(∏ B_{i}).

To prove the result for Π/Σ, we note that

\xymatrix{0 \ar[r] & K_{∗}(Σ) \ar[r] & K_{∗}(Π) \ar[r] & K_{∗}(Π/Σ) \ar[r] & 0}

is exact and apply the argument above to a x ∈ K_{0}(Π) lifting the given element in K_{0}(Π/Σ). ■

Algebraically compact K-groups

An abelian group G is algebraically compact when Pext(−, G) vanishes. This class of groups is well studied (cf. [24, VII]), and we are going to use the characterization of it as those groups for which

(i) The subgroup ⋂_{n∈ℕ} nG is divisible. (ii) G is complete hold ([26]). Here completeness refers to the ℤ-adic topology (cf. [24, 7]), and completeness does not (as it does in [24]!) imply any separation properties.

Note that the quantities pfo and ipo make sense for general ordered abelian groups with order unit. We extend the notions to such groups and families of them in the obvious way. When (G_{i}, 1_{i}) is a family of groups and order units, we set

∏_{b} G_{i} = {(g_{i})| ∃M ∈ ℕ ∀i ∈ I : −M1_{i} ≤ g_{i} ≤ M1_{i}} .

Lemma 3.5 Whenever G_{i} is a sequence of abelian groups, then ∏ G_{i}/ ∑ G_{i} is algebraically compact. If, furthermore, all G_{i} are ordered with order units, then ∏_{b} G_{i}/ ∑ G_{i} is algebraically compact provided that both pfo({G_{i}}) and ipo({G_{i}}) are finite.

Proof: When X = ∏ G_{i}/ ∑ G_{i}, then X is algebraically compact by [27]. When the G_{i} are also ordered, let X_{b} = ∏_{b} G_{i}/ ∑ G_{i} and consider X_{b} as a subgroup of X. We are going to prove that (i) and (ii) above hold for X_{b} from the fact that they hold for X.

Since pfo({G_{i}}) < ∞, we immediately get that if x ∈ X_{b} and x = my in X, then y ∈ X_{b} if m ≠ 0, by an argument very similar to the one at the end of proof of Lemma 3.2. Note that this is stronger than mere purity of X_{b} as a subgroup of X. Fix m ≠ 0 and x ∈ ⋂_{n∈ℕ} nX_{b} and write x = my for y ∈ ⋂_{n∈ℕ} nX. Applying this observation twice, we get

y ∈ X_{b} ∩ ⋂_{n∈ℕ}nX = ⋂_{n∈ℕ}(nX ∩ X_{b}) = ⋂_{n∈ℕ}nX_{b}.

It remains to show that X_{b} is complete. To do so, we first note that for any given x ∈ X there is y ∈ X_{b} such that x − y ∈ ⋂_{n∈ℕ} nX. For when ipo({G_{i}}) ≤ ℓ and x = (x_{i}) + ∑ G_{i} is given, we may find y_{i} such that −ℓ1_{i} ≤ y_{i} ≤ ℓ1_{i} and x_{i} − y_{i} ∈ i!G_{i}. Clearly y = (y_{i}) + ∑ G_{i} has the desired property. Now let (x_{n}) be Cauchy in X_{b}, and recall that it converges to some x ∈ X. With y chosen as above, (x_{n}) also converges to y ∈ X_{b} in the ℤ-adic topology of X_{b}. This is because the ℤ-adic topology of X_{b} coincides with the topology of X_{b} induced by the ℤ-adic topology of X, since X_{b} is a pure subgroup of X. ■

Corollary 3.6 Let B_{i} be a sequence of unital C^{∗}-algebras.

(i) K_{1}(∏ B_{i}/ ∑ B_{i}) is algebraically compact if rr({B_{i}}) = 0 and elo_{1}({B_{i}}) < ∞. (ii) K_{0}(∏ B_{i}/ ∑ B_{i}) is algebraically compact if either cco({B_{i}}), elo_{0}({B_{i}}) < ∞

or cco({B_{i}}), pfo({B_{i}}), ipo({B_{i}}) < ∞.

Proof: Part (i) follows from Lemmas 3.3 and 3.5. Part (ii) follows from Lemmas 3.2 and 3.5. ■

3.3 Partial maps on \underline{K}(-)

In this section we concern ourselves with associating K-theoretical data to completely positive contractions. Starting from such maps, say from A to B, we shall be able to induce maps sending a finite set of projections representing a finite part of the K-theory of A to elements of the K-theory of B.

Although there are advantages of doing this even for subsets representing elements of K_{0}(A), the real strength of this approach only surfaces when we work with all of \underline{K} (A) and \underline{K} (B). Our partial maps do not descend to well-defined maps on subsets of \underline{K} (A), let alone to all of \underline{K} (A), but this fact does not cause any problems except notational and technical inconveniences.

As noted in [15] we can realize any element of \underline{K} (−) as a difference of projections from

\underline{Proj}(A) = ⋃_{m≥1} Proj(A ⊗ C(𝕋) ⊗ C(W_{m}) ⊗ K)

where the W_{m} are the Moore spaces of order m. This picture of \underline{K} (A) encompasses the standard pictures of K_{0}(A) and K_{∗}(A) using projections of A ⊗ K and A ⊗ C(𝕋) ⊗ K, respectively, but not the standard picture of K_{1}(A) using unitaries of (A ⊗ K)^{∼}. We need to pay special attention to this. Checking the facts stated as lemmas below is tedious but straight-forward. We leave it to the reader with due apologies.

Definition 3.7 Let A be a C^{∗}-algebra. A \underline{K}-triple (P, G, δ) consists of finite subsets P ⊆ \underline{Proj}(A) and G ⊆ A and a δ > 0 chosen such that whenever φ is a completely positive contraction which is δ-multiplicative on G, then

\frac{1}{2} ∉ sp((φ ⊗ id)(p))

for each p ∈ P, where id is the identity of C(𝕋)⊗C(W_{m})⊗K for suitable m. A K_{1}-triple (V, G, δ) consists of finite subsets V ⊆ U((A ⊗ K)^{∼}) and G ⊆ A and a δ > 0 chosen such that whenever φ is a completely positive contraction which is δ-multiplicative on G, then

0 ∉ sp((φ ⊗ id)(v))

for each v ∈ V.

We define K_{0}- and K_{∗}-triples analogously to the \underline{K}-triple case, by using projections in A ⊗ K and A ⊗ C(𝕋) ⊗ K, respectively.

The following lemma shows that any finite subset of projections or unitaries can be complemented to a triple of the appropriate kind.

Lemma 3.8 Let A and C be C^{∗}-algebras, and fix finite sets P, V ⊆ (A⊗C)^{∼} consisting of projections and unitaries, respectively. Then there exists δ > 0 and a finite set G ⊆ A such that whenever φ : A ⟶ B is a unital completely positive map which is δ-multiplicative on G, then

\frac{1}{2} ∉ sp((φ ⊗ id_{C})(p)) 0 ∉ sp((φ ⊗ id_{C})(v))

for every p ∈ P, v ∈ V.

Let χ_{0} : [0, \frac{1}{2})∪(\frac{1}{2}, 1] → [0, 1] be 0 on [0, \frac{1}{2}) and 1 on (\frac{1}{2}, 1], and let χ_{1} : (0, 1] → [0, ∞) be given by χ_{1}(x) = x^{−1/2}.

Definition 3.9 Let (P, G, δ) be a \underline{K}-triple, and assume that φ : A ⟶ B is a completely positive contraction which is δ-multiplicative on G. We define φ_{♯} : P ⟶ \underline{K} (B) by

φ_{♯}(p) = [χ_{0}(φ ⊗ id)(p)].

When (V, G, δ) is a K_{1}-triple, we define φ_{♯} : V → K_{1}(B) by

φ_{♯}(v) = [V χ_{1}(V V^{∗}))]

where V = (φ ⊗ id)(v).

Maps into K_{0}(B) and K_{∗}(B) are defined from K_{0}-triples and K_{∗}-triples similarly to the \underline{K}-triple case.

Lemma 3.10 Let A be a unital C^{∗}-algebra and (P, G, δ) a \underline{K}-triple. Let φ : A ⟶ B be a completely positive map which is δ-multiplicative on G and let j : B ⟶ C be a unital ∗-homomorphism. Then (jφ)_{♯}(p) = j_{∗}φ_{♯}(p) for all p ∈ P.

To establish the next two results, one may use that the canonical map from K_{0}(A) to K_{1}(SA) is defined using scalar rotation matrices. The last result follows from the definitions.

Lemma 3.11 Whenever a K_{∗}-triple (P_{∗}, G, δ) is given, there exist a K_{0}-triple (P_{0}, G_{0}, δ_{0}) and a K_{1}-triple (V, G_{1}, δ_{1}) with δ_{i} < δ and G_{i} ⊇ G such that if for two unital completely positive contractions φ, ψ which are δ_{i}-multiplicative on G_{i},

φ_{♯}(p) = ψ_{♯}(p) φ_{♯}(v) = ψ_{♯}(v)

for all p ∈ P_{0} and all v ∈ V, then φ_{♯}(p) = ψ_{♯}(p) for all p ∈ P_{∗}.

Lemma 3.12 Whenever a K_{0}-triple (P_{0}, G_{0}, δ_{0}) and a K_{1}-triple (V, G_{1}, δ_{1}) is given, there exists a K_{∗}-triple (P_{∗}, G, δ) with δ < δ_{i} and G ⊇ G_{i} such that if for two unital completely positive contractions φ, ψ which are δ-multiplicative on G, φ_{♯}(p) = ψ_{♯}(p) for all p ∈ P_{∗}, then

φ_{♯}(p) = ψ_{♯}(p) φ_{♯}(v) = ψ_{♯}(v)

for all p ∈ P_{0} and all v ∈ V.

Lemma 3.13 Let (P, G, δ) be a \underline{K}-triple on A. There exists ε > 0 and a finite set F ⊆ A such that if φ, ψ are completely positive contractions which are δ-multiplicative on G and satisfy ∥φ(a) − ψ(a)∥ < ε for all a ∈ F, then φ_{♯}(p) = ψ_{♯}(p) for all p ∈ P.

4 Uniqueness

In the present section we present our uniqueness results as a sequence of successive refinements, at the cost of added assumptions.

We summarize our results in the following table; the reader should be able to deduce the precise meaning of the various columns by comparing with the actual statements of the uniqueness results below. In the table we require tacitly that every A is a separable C^{∗}-algebra and every map φ, ψ is nuclear.

Ref. A  B  (A, B)  φ, ψ  ⊕
4.4  ∃γ : A ⟶ M(K ⊗ B) morphisms γ
nuclearly absorbing  [φ] = [ψ]
4.5 unital unital ∃γ : A ⟶ M(K ⊗ B) morphisms γ_{n}
unital  [φ] = [ψ]
nuclearly absorbing  φ(1) ∼ ψ(1)

quasidiagonal

4.12 unital unital ∃ι : A ↪ B  morphisms n · ι
simple 4.9 unital morphism φ_{∗} = ψ_{∗}
exact  φ(1) ∼ ψ(1)

UCT

4.15 unital unital ∃τ : A ↪ B  CP-maps  n · τ
simple 4.9  unital CP-map  φ_{♯}|P = ψ_{♯}|P
exact  φ(1) ∼ ψ(1)

UCT

4.1 Stable uniqueness

We repeat the following definition and main result from [14]:

Definition 4.1 If π, σ : A → L(E) are representations, we say that π and σ are properly asymptotically unitarily equivalent and write π ≊ σ if there is a continuous path of unitaries u : [0, ∞) → U(K(E) + ℂ1_{E}), u = (u_{t})_{t∈[0,∞)} such that for all a ∈ A,

(i) lim_{t→∞} ∥u_{t}π(a)u_{t}^{∗} − σ(a)∥ = 0,

(ii) u_{t}π(a)u_{t}^{∗} − σ(a) ∈ K(E), for all t ∈ [0, ∞).

Theorem 4.2 Let A be a (unital) separable C^{∗}-algebra and let B be a σ-unital C^{∗}- algebra. Let (φ, ψ) : A → M(K(H) ⊗ B) be a Cuntz pair consisting of (unital) strictly nuclear representations. Then the following are equivalent:

(i) [φ, ψ] = 0 in KK_{nuc}(A, B)

(ii) There exists a (unital) strictly nuclear representation σ : A → M(K(H) ⊗ B) with φ ⊕ σ ≊ ψ ⊕ σ (iii) For any (unital) nuclearly absorbing representation γ : A → M(K(H) ⊗ B), φ ⊕ γ_{∞} ≊ ψ ⊕ γ_{∞} (iv) For any (unital) nuclearly absorbing and strictly nuclear representation γ : A → M(K(H) ⊗ B), φ ⊕ γ_{∞} ≊ ψ ⊕ γ_{∞}

It is proved in [14] that

φ ⊕ σ ≊ ψ ⊕ σ ⟺ φ ⊕ γ ≊ ψ ⊕ γ

if σ and γ are asymptotically equivalent in a sense made precise in that paper. The same proof shows the following:

Lemma 4.3 Let φ, ψ, γ, σ : A → M(K(H) ⊗ B) be unital representations. If φ ⊕ γ ≊ ψ ⊕ γ, and γ ∼ σ, then there exists a sequence of unitaries v_{n} ∈ K(H) + ℂ1 such that

∥v_{n}(φ ⊕ σ)(a)v_{n}^{∗} − (ψ ⊕ σ)(a)∥ ⟶ 0 n ⟶ ∞

for any a ∈ A.

The previous lemma simplifies matters when working with uniqueness — it is necessary to work with asymptotic equivalence of cycles to characterize KK-theory as above, but to prove uniqueness results one only needs an approximate version of (i)⟹(iii) above. Hence we get:

Corollary 4.4 Let A be a separable C*-algebra and let B be a σ-unital C^{∗}-algebra. Assume that φ and ψ are two nuclear ∗-homomorphisms from A to K(H) ⊗ B which satisfy [φ] = [ψ] in KK_{nuc}(A, B). Let γ : A ⟶ M(K(H) ⊗ B) be a nuclearly absorbing representation. Whenever a finite subset F ⊆ A and ε > 0 is given, there exists a unitary v ∈ M_{3}(K(H)) ⊗ B + ℂ1 such that

∥v [\begin{matrix}φ(a) \\ & 0 \\ & & γ(a) \end{matrix}] v^{∗} − [\begin{matrix}ψ(a) \\ & 0 \\ & & γ(a) \end{matrix}]∥ < ε

for all a ∈ F.

Proof: The corollary follows from the non-unital version of Theorem 4.2 after the following remarks. Since φ and ψ assume only compact values, we have

[φ, ψ] = [φ, 0] − [ψ, 0] = 0

in KK_{nuc}(A, B). Also γ̂ is (nonunital) nuclearly absorbing by Lemma 2.17, whether or not γ is unital. We conclude that φ ⊕ γ̂_{∞} ≊ ψ ⊕ γ̂_{∞} and, since γ̂ ∼ γ̂_{∞}, that v exists by Lemma 4.3. ■

Achieving stable uniqueness

We are now going to employ the quasidiagonality condition 2.5 to truncate the absorbing representation in Corollary 4.4 to something more manageable in terms of classification.

We consider two ∗-homomorphisms φ and ψ, between unital C^{∗}-algebras A and B. To fix notation, let γ : A ⟶ M(K(H) ⊗ B) be a quasidiagonal unital representation. We consider a quasidiagonalization (γ_{n}) : A ⟶ M_{r_{n}}(B) by (e_{n}) (see the paragraph following 2.5), where we may and shall assume that (e_{n}) has the additional property that

e_{n}φ(a)e_{n} = φ(a) e_{n}ψ(a)e_{n} = ψ(a), n ≥ 1.

Theorem 4.5 Let A be a unital, separable C^{∗}-algebra and let B be a unital C^{∗}- algebra. Assume that there exists a unital quasidiagonal nuclearly absorbing representation γ : A ⟶ M(K(H) ⊗ B), and let (γ_{n}) : A ⟶ M_{r_{n}}(B) be a quasidiagonalization of γ by (e_{n}) as above.

Suppose that φ, ψ : A ⟶ B are two nuclear ∗-homomorphisms with [φ] = [ψ] in KK_{nuc}(A, B), such that φ(1) is unitarily equivalent to ψ(1). Then for any finite subset F ⊆ A and any ε > 0, there exist an integer n and a unitary u ∈ U_{r_{n}+1}(B) satisfying

∥u [\begin{matrix}φ(a) \\ & γ_{n}(a) \end{matrix}] u^{∗} − [\begin{matrix}ψ(a) \\ & γ_{n}(a) \end{matrix}]∥ < ε

for all a ∈ F. Moreover we may arrange that u(φ(1) ⊕ γ_{n}(1))u^{∗} = ψ(1) ⊕ γ_{n}(1).

Proof: After conjugating ψ by a unitary in B we may assume that φ(1) = ψ(1). We are going to compress by e′_{n} = e_{n} ⊕ e_{n} ⊕ e_{n} (which is a quasicentral sequence in K(H_{3}) ⊗ B + ℂ1_{3}), in the conclusion of Corollary 4.4. We have

e′_{n} [ \begin{matrix}φ(a) \\ & 0 \\ & & γ(a) \end{matrix} ] e′_{n} = [ \begin{matrix}φ(a) \\ & 0 \\ & & γ_{n}(a) \end{matrix} ]

and a similar equation for ψ. It is crucial to our argument that the unitary v provided by Corollary 4.4 satisfies ∥[v, e′_{n}]∥ → 0 because v ∈ K(H_{3}) ⊗ B + ℂ1_{3}. Therefore by perturbing e′_{n}ve′_{n} to a unitary z within U_{3r_{n}}(B), for some large n, we obtain:

∥z [ \begin{matrix}φ(a) \\ & 0 \\ & & γ_{n}(a) \end{matrix} ] z^{∗} − [ \begin{matrix}ψ(a) \\ & 0 \\ & & γ_{n}(a) \end{matrix} ]∥ < ε

for all a ∈ F. Consider the projection e = φ(1) ⊕ γ_{n}(1) = ψ(1) ⊕ γ_{n}(1). After a small perturbation of z we may assume that zez^{∗} = e. Then w = eze is partial isometry in M_{r_{n}+1}(B) with w^{∗}w = ww^{∗} = e, and the unitary u = w + 1_{r_{n}+1} − e ∈ U_{r_{n}+1}(B) will satisfy the conclusion of the theorem. ■

The following corollary is a good illustration of Theorem 4.5.

Corollary 4.6 Let A be a separable unital residually finite dimensional C^{∗}-algebra and let B be a unital C^{∗}-algebra. Let (π_{n}) be a separating sequence of finite dimensional representations of A such that each π_{n} is repeated infinitely many times. Define γ_{n} = π_{1} ⊕ ⋯ ⊕ π_{n}, with γ_{n} : A → M_{r_{n}}(ℂ) ⊆ M_{r_{n}}(ℂ1_{B}). Let φ, ψ : A → B be two unital nuclear ∗-homomorphisms with [φ] = [ψ] in KK_{nuc}(A, B). Then there is a sequence of unitaries u_{n} ∈ U_{r_{n}+1}(B) such that

lim_{n→∞} ∥u_{n}(φ(a) ⊕ γ_{n}(a))u_{n}^{∗} − ψ(a) ⊕ γ_{n}(a)∥ = 0, a ∈ A.

Proof: This follows from Theorem 4.5 with γ = π_{1} ⊕ π_{2} ⊕ · · ·. Note that since γ is an admissible scalar representation, γ is nuclearly absorbing by Proposition 2.18. ■

4.2 Some classes of C^{∗}-algebras

We single out several classes of C^{∗}-algebras for easy reference:

Definition 4.7 We say that a separable C^{∗}-algebra A satisfies the UCT if the diagram

\xymatrix{0 \ar[r] & Ext(K_{∗}(A), K_{∗+1}(B)) \ar[r] & KK(A, B) \ar[r] & Hom(K_{∗}(A), K_{∗}(B)) \ar[r] & 0}

is a short exact sequence for every σ-unital algebra B.

A large class of algebras satisfying UCT was exhibited in [47]. If the separable C^{∗}-algebra A satisfies the UCT, then for any σ-unital C^{∗}-algebra B the sequence:

\xymatrix{0 \ar[r] & Pext(K_{∗}(A), K_{∗+1}(B)) \ar[r] & KK(A, B) \ar[r] & Hom_{Λ}(\underline{K} (A) , \underline{K} (B)) \ar[r] & 0} (4.1)

is also exact by [18].

Remark 4.8 It is not known whether there exist separable nuclear C^{∗}-algebras not satisfying the UCT. If a separable C^{∗}-algebra A satisfies the UCT, then A is KKequivalent to a commutative C^{∗}-algebra, hence it is K-nuclear. In particular KK_{nuc}(A, B) = KK(A, B) by [51].

Definition 4.9 A C^{∗}-algebra B is called an admissible target algebra if it is unital, has real rank zero ([5]) and satisfies (i) whenever p, q ∈ Proj(B ⊗ K), then [p] = [q] ⟹ p ⊕ 1_{B} ∼ q ⊕ 1_{B} (ii) the canonical map U_{1}(B) ⟶ K_{1}(B) is surjective

and if either

(iii.∞) the canonical map Proj(B) ⟶ K_{0}(B) is surjective

or both of

(iii.1) For any x ∈ K_{0}(B) such that nx ≥ 0 for some n ≥ 0, one has x + [1_{B}] ≥ 0. (iv.1) For any x ∈ K_{0}(B) and any n ≠ 0, there is y ∈ K_{0}(B) such that −[1_{B}] ≤

y ≤ [1_{B}] and x − y ∈ nK_{0}(B).

holds.

When needed, we distinguish between admissible target algebras satisfying (iii.1)– (iv.1) or (iii.∞) by calling them admissible of finite type or infinite type, respectively. Examples will be given in Propositions 6.6 and 6.16.

The point of this definition is that whenever a sequence of admissible targets are given, then both of the natural maps

(4.2) Hom_{Λ}(\underline{K} (A) , \underline{K} (∏ B_{i})) ⟶ ∏ Hom_{Λ}(\underline{K} (A) , \underline{K} (B_{i})) KK(A, ∏ B_{i}/ ∑ B_{i}) ⟶ Hom_{Λ}(\underline{K} (A) , \underline{K} (∏ B_{i}/ ∑ B_{i})) (4.3)

will be injective (the latter in fact an isomorphism) for an A satisfying the UCT:

Theorem 4.10 Let B_{i} be a sequence of admissible target algebras of the same type and let A be any C^{∗}-algebra. Then

(i) \underline{ηηη} : \underline{K} (∏ B_{i}) ⟶ ∏ \underline{K} (B_{i}) is injective.

(ii) The natural map Hom_{Λ}(\underline{K} (A) , \underline{K} (∏ B_{i})) ⟶ ∏ Hom_{Λ}(\underline{K} (A) , \underline{K} (B_{i})) is injective. (iii) The natural map KK(A, ∏ B_{i}/ ∑ B_{i}) ⟶ Hom_{Λ}(\underline{K} (A) , \underline{K} (∏ B_{i}/ ∑ B_{i})) is an isomorphism if A satisfies the UCT.

(iv) ∏ B_{i} and ∏ B_{i}/ ∑ B_{i} are admissible targets

Proof: For (i), decompose \underline{ηηη} into maps η^{∗} and η_{n}^{∗}. We get from Lemma 3.2 and 3.3 that η^{∗} are injections whose images are pure subgroups. Injectivity of η_{n}^{∗} then follows by a diagram chase on

\xymatrix{K_{∗}(∏ B_{i}) \ar[r]^{×n} \ar[d]_{η^{∗}} & K_{∗}(∏ B_{i}) \ar[r]^{ρ_{n}^{∗}} \ar[d]_{η^{∗}} & K_{∗}(∏ B_{i}; ℤ/n) \ar[r]^{β_{n}^{∗}} \ar[d]_{η_{n}^{∗}} & K_{∗+1}(∏ B_{i}) \ar[d]_{η^{∗+1}} \\ _{∏ K_{∗}(B_{i})} \ar[r]_{×n} & _{∏ K_{∗}(B_{i})} \ar[r]_{∏ ρ_{n}^{∗}} & _{∏ K_{∗}(B_{i}; ℤ/n)} \ar[r]_{∏ β_{n}^{∗}} & _{∏ K_{∗+1}(B_{i}).}}

Claim (ii) is a direct consequence of (i), and (iii) follows by combining the UMCT (4.1) with Corollary 3.6. Finally, (iv) follows by Lemma 3.4. ■

Remark 4.11 As is clear from Section 3.2, we can get injectivity for the maps discussed above asserting considerably weaker versions of (iii)-(iv). On the other hand, B_{i} = C([0, 1]) is a counterexample to injectivity in (4.2) and B_{i} = ℂ a counterexample to (4.3).

4.3 Stable uniqueness with bounds

Let A be a separable unital C^{∗}-algebra and let ι : A → B be a unital full embedding. If one specializes to the case γ = d_{ι} in Theorem 4.5, one obtains a stable approximate unitary equivalence of the form

∥u [\begin{matrix}φ(a) \\ & n·ι(a) \end{matrix}] u^{∗} − [\begin{matrix}ψ(a) \\ & n·ι(a) \end{matrix}]∥ < ε.

To make such a result useful in our quest to classify C^{∗}-algebras, we need to refine our uniqueness results to the effect of controlling the number n. More specifically, we need to know that these integers can be chosen uniformly with respect to the targets; i.e. only depending on the source algebra and, of course, the requirements on how closely the two morphisms are to agree after composition by the unitary.

We also need to strengthen the theorem to allow for maps which are not ∗-homomorphisms and only induce the same element locally in Hom_{Λ}(\underline{K} (A) , \underline{K} (B)), rather than in KK(A, B). To achieve such results, we are going to work with products of C^{∗}-algebras, and we are going to depend on the results in Section 3.2 regarding their K-theory. If A satisfies the UCT, then if follows from (4.1) that Hom_{Λ}(\underline{K} (A) , \underline{K} (B)) is isomorphic to Rørdam’s group KL(A, B) [46].

Bounded stable uniqueness for ∗-homomorphisms

Theorem 4.12 Let A be a simple, unital, exact, separable C^{∗}-algebra satisfying the UCT. Then for any finite subset F ⊆ A and any ε > 0, there exists n ∈ ℕ with the following property. For any admissible target B, any unital embedding ι : A ⟶ B and any pair of nuclear ∗-homomorphisms φ, ψ : A ⟶ B such that φ_{∗} = ψ_{∗} in Hom_{Λ}(\underline{K} (A) , \underline{K} (B)), and φ(1) is unitarily equivalent to ψ(1), there exists a unitary u ∈ U_{n+1}(B) such that

∥u [\begin{matrix}φ(a) \\ & n·ι(a) \end{matrix}] u^{∗} − [\begin{matrix}ψ(a) \\ & n·ι(a) \end{matrix}]∥ < ε.

for all a ∈ F. Moreover we may arrange that u(φ(1) ⊕ n·1)u^{∗} = ψ(1) ⊕ n·1.

Proof: We have KK_{nuc}(A, B) = KK(A, B) by Remark 4.8. Seeking a contradiction, fix F and ε for which the theorem fails. Then for any i we choose an admissible target algebra B_{i} equipped with an embedding ι_{i} : A ⟶ B_{i}, and φ_{i}, ψ_{i} nuclear ∗- homomorphisms with φ_{i}_{∗} = ψ_{i}_{∗}, and φ_{i}(1) unitarily equivalent to ψ_{i}(1), yet

inf_{u∈U_{i+1}(B)} max_{a∈F} ∥u [\begin{matrix}φ_{i}(a) \\ & i·ι_{i}(a) \end{matrix}] u^{∗} − [\begin{matrix}ψ_{i}(a) \\ & i·ι_{i}(a) \end{matrix}]∥ ≥ ε.

We define Φ, Ψ, I : A ⟶ ∏ B_{i} in the obvious way. Φ and Ψ are nuclear since φ_{i}, ψ_{i} are nuclear and A is exact (see [13, 3.3]). By composing with the canonical map we get Φ̇, Ψ̇, İ : A ⟶ ∏ B_{i}/ ∑ B_{i}, with Φ̇, Ψ̇ nuclear. Since Φ and Ψ induce the same element of ∏ Hom_{Λ}(\underline{K} (A) , \underline{K} (B_{i})) by construction, we get from Theorem 4.10(ii) that Φ_{∗} = Ψ_{∗} in Hom_{Λ}(\underline{K} (A) , \underline{K} (∏ B_{i})). Then of course also (Φ̇)_{∗} = (Ψ̇)_{∗}, and by Theorem 4.10(iii) we get that [Φ̇]_{KK} = [Ψ̇]_{KK} in KK(A, ∏ B_{i}/ ∑ B_{i}).

Since İ : A ⟶ ∏ B_{i}/ ∑ B_{i} is a unital full embedding, we conclude by Theorem 4.5 that there exist n and a unitary w ∈ U_{n+1}(∏ B_{i}/ ∑ B_{i}) intertwining Φ̇ ⊕ n·İ and Ψ̇ ⊕ n·İ up to ε on F. Let u = (u_{i}) ∈ U_{n+1}(∏ B_{i}) be a unitary lifting w. Then

lim sup_{i} max_{a∈F} ∥u_{i} [\begin{matrix}φ_{i}(a) \\ & n·ι_{i}(a) \end{matrix}] u_{i}^{∗} − [\begin{matrix}ψ_{i}(a) \\ & n·ι_{i}(a) \end{matrix}]∥ < ε.

yielding a contradiction after projecting onto M_{n+1}(B_{i}) for large i. The last part of the proof is done exactly as the last part of the proof of Theorem 4.5. ■ Remark 4.13 If φ and ψ are as in the conclusion of either Theorem 4.5 or Theorem 4.12, it follows immediately from the cancellative nature of K-theory that φ_{∗} = ψ_{∗} : \underline{K} (A) → \underline{K} (B). Remark 4.14 Under assumptions restricting the algebraic complexity on K_{∗}(A) and K_{∗}(B) the result above can be simplified somewhat. If we add, for instance, the assumptions that K_{0}(A) be torsion free and K_{0}(B) be divisible, we need only require that φ_{∗} = ψ_{∗} on K_{∗}(A). This is done basing the proof instead on injectivity of the maps

Hom(K_{∗}(A), K_{∗}(∏ B_{i})) ⟶ ∏ Hom(K_{∗}(A), K_{∗}(B_{i}))
KK(A, ∏ B_{i}/ ∑ B_{i}) ⟶ Hom(K_{∗}(A), K_{∗}(∏ B_{i}/ ∑ B_{i})).

We get the latter by applying the UCT and noting that since K_{0}(A) is torsion free

Ext (K_{0}(A), K_{1} (∏ B_{i}/ ∑ B_{i})) = Pext (K_{0}(A), K_{1} (∏ B_{i}/ ∑ B_{i})) = 0 from Corollary 3.6(i), and that

Ext (K_{1}(A), K_{0} (∏ B_{i}/ ∑ B_{i})) = 0

since, along the lines of the first half of Lemma 3.5, if all K_{0}(B_{i}) are divisible, then so is K_{0}(∏ B_{i}).

Bounded stable uniqueness for approximate morphisms

We refer the reader to Section 3.3 for a discussion of partially defined maps on \underline{K} (−) and a definition of \underline{K}-triples.

Theorem 4.15 Let A be a simple, unital, exact, separable C^{∗}-algebra satisfying the UCT. For any finite subset F ⊆ A and any ε > 0, there exist n ∈ ℕ, and a \underline{K}-triple (P, G, δ) with the following property. For any admissible target B, and any three completely positive contractions φ, ψ, τ : A ⟶ B which are δ-multiplicative on G, with τ unital, φ, ψ nuclear and φ_{♯}(p) = ψ_{♯}(p) in \underline{K} (B) for all p ∈ P, and such that φ(1) and ψ(1) are unitarily equivalent projections, there exists a unitary u ∈ U_{n+1}(B) such that

∥u [\begin{matrix}φ(a) \\ & n·τ(a) \end{matrix}] u^{∗} − [\begin{matrix}ψ(a) \\ & n·τ(a) \end{matrix}]∥ < ε

for all a ∈ F. One may arrange that u(φ(1) ⊕ n·1)u^{∗} = ψ(1) ⊕ n·1.

Proof: We have KK_{nuc}(A, B) = KK(A, B) by Remark 4.8. Seeking a contradiction we suppose that there is F and ε such that with n = n(A, F, ε) provided by Theorem 4.12, no \underline{K}-triple will work. We choose sequences of \underline{K}-triples (P_{i}, G_{i}, δ_{i}) with the properties

(i) P_{i} ⊆ P_{i+1}, and ⋃_{i∈ℕ} P_{i} exhausts the semigroup

⋃_{m∈ℕ}Proj(A ⊗ C(𝕋) ⊗ C(W_{m}) ⊗ K)/ ≈

(ii) G_{i} ⊆ G_{i+1}, \overline{⋃_{i} G_{i}} = A. (iii) δ_{i} > δ_{i+1}, δ_{i} ⟶ 0.

By our assumption, we can then choose admissible targets B_{i}, which we may assume are of the same type, and φ_{i}, ψ_{i} and τ_{i} which are δ_{i}-multiplicative on G_{i}, with φ_{i}, ψ_{i} nuclear, and satisfy (φ_{i})_{♯}(p) = (ψ_{i})_{♯}(p) in \underline{K} (B) for all p ∈ P, and φ_{i}(1), ψ_{i}(1) are unitarily equivalent projections; yet

inf_{u∈U_{n+1}(B)} max_{a∈F} ∥u [ \begin{matrix}φ_{i}(a) \\ & n·τ_{i}(a) \end{matrix} ] u^{∗} − [ \begin{matrix}ψ_{i}(a) \\ & n·τ_{i}(a) \end{matrix} ]∥ ≥ ε.

Define Φ, Ψ, T : A ⟶ ∏ B_{i} from the sequences (φ_{i}), (ψ_{i}) and (τ_{i}), and compose with the canonical map to get Φ̇, Ψ̇, Ṫ : A ⟶ ∏ B_{i}/ ∑ B_{i}. These maps are in fact ∗- homomorphisms by (ii) and (iii) above, so Ṫ provides a unital full embedding of A into ∏ B_{i}/∑ B_{i}, and Φ̇, Ψ̇ induce maps

Φ̇_{∗}, Ψ̇_{∗} : \underline{K} (A) ⟶ \underline{K} (∏ B_{i}/ ∑ B_{i}) .

We are going to show that Φ̇_{∗} = Ψ̇_{∗}.

We may check this on p ∈ P_{j} by (i) above. Let C = C(𝕋) ⊗ C(W_{m}) ⊗ K, with m chosen appropriately. In the diagram

\xymatrix{0 \ar[r] & K_{0} ((∑ B_{i}) ⊗ C) \ar[r] \ar[d]^{≅} & K_{0} ((∏ B_{i}) ⊗ C) \ar[r] \ar[d]^{η} & K_{0} (\begin{matrix}∏ & B_{i} \\ ∑ & B_{i} \end{matrix} ⊗ C) \ar[r] \ar[d]^{η̇} & 0 \\ 0 \ar[r] & ∑ K_{0} (B_{i} ⊗ C) \ar[r] & ∏ K_{0} (B_{i} ⊗ C) \ar[r]|{κ} & \frac{∏ K_{0} (B_{i} ⊗ C)}{∑ K_{0} (B_{i} ⊗ C)} \ar[r] & 0}

η is injective from Theorem 4.10(i) because it can be naturally identified with a component of \underline{ηηη}. It is not hard to check that the above diagram has exact rows. (The first row is induced by a quasidiagonal extension). By the five-lemma, η̇ is also injective, and hence it suffices to show that

η̇[(Φ̇ ⊗ id_{C})(p)] = η̇[(Ψ̇ ⊗ id_{C})(p)].

With χ_{0} as in Section 3.3, lift (Φ̇ ⊗ id)(p) first to a self-adjoint element (Φ ⊗ id)(p) in (∏ B_{i}) ⊗ C, and then to a projection q in

χ_{0}((Φ ⊗ id)(p)) + (∑ B_{i}) ⊗ C ⊆ (∏ B_{i}) ⊗ C.

Then η([q]) = ([q_{i}]) where q_{i} = χ_{0}((φ_{i} ⊗ id)(p)) for all i larger than some i_{Φ}, since η is induced by a family of ∗-homomorphisms. We conclude that

η̇((Φ̇ ⊗ id)_{∗}(p)) = κη([q]) = [χ_{0}((φ_{i} ⊗ id)(p))]_{i≥i_{Φ}} + ∑ K_{0}(B_{i} ⊗ C).

Similarly,

η̇((Ψ̇ ⊗ id)_{∗}(p)) = [χ_{0}((ψ_{i} ⊗ id)(p))]_{i≥i_{Ψ}} + ∑ K_{0}(B_{i} ⊗ C),

and these elements agree since the sequences coincide for i ≥ i_{Φ}, i_{Ψ}, j.

Note that Φ and Ψ, and hence also Φ̇ and Ψ̇, are nuclear since φ_{i} and ψ_{i} are nuclear and A is exact [13]. Having proved that Φ̇_{∗} = Ψ̇_{∗}, and since Φ̇(1) is unitarily equivalent to Ψ̇(1), we may thus apply Theorem 4.12 to find a unitary w ∈ U_{n+1}(∏ B_{i}/ ∑ B_{i}) such that

∥w [ \begin{matrix}Φ̇(a) \\ & n·Ṫ (a) \end{matrix} ] w^{∗} − [ \begin{matrix}Ψ̇(a) \\ & n·Ṫ (a) \end{matrix} ]∥ < ε.

for all a ∈ F. Note that we may apply Theorem 4.12 since whenever B_{i} is a sequence of admissible targets, ∏ B_{i}/ ∑ B_{i} is an admissible target by Theorem 4.10(iv). We finish the argument by lifting the unitary as above. ■

Remark 4.16 As in Remark 4.14, the premises of Theorem 4.15 simplify under extra assumptions on K_{∗}(A) and K_{∗}(B). If K_{0}(A) is torsion free and K_{0}(B) is divisible, one needs only check that φ and ψ agree on a K_{∗}-triple. This is because divisibility passes from K_{0}(B_{i}) to K_{0}(∏ B_{i}/ ∑ B_{i}) as outlined in Remark 4.14.

5 Existence

In this section we show how to realize locally a given KK-element by a difference of completely positive contractions (see Theorem 5.5).

5.1 Realizing group homomorphisms

We refer the reader to Section 3.3 for the definition of \underline{K}-triples. Let χ_{0} be the characteristic function of (1/2, 1].

Lemma 5.1 Let D be a unital C^{∗}-algebra. Let p ∈ D be a projection and let x ∈ D be a positive contraction.

(i) |x − \frac{1}{2}| ≥ \frac{1}{2} − ∥x − p∥.

(ii) sp(x) ⊂ [0, λ] ∪ [1 − λ, 1], where λ is the smaller root of λ^{2} − λ + ∥x^{2} − x∥ = 0. Thus if ∥x^{2} − x∥ < \frac{1}{4}, then χ_{0}(x) ∈ D is a projection with ∥x − χ_{0}(x)∥ ≤ λ. (iii) If (f_{n}) is a sequence of projections such that [p, f_{n}] → 0 as n → ∞, then there is a sequence of unitaries (v_{n}) in D, such that [v_{n}p v_{n}^{∗}, f_{n}] = 0, n ≥ 1 and ∥v_{n} − 1∥ → 0 as n → ∞.

Proof: For (i), we may assume that D acts on a Hilbert space H, for (ii), that D = C(X) with X compact. After these assumptions the claims become elementary. In (iii), we may assume that ∥[p, f_{n}]∥ < 1/4 for all n. Define q_{n} = χ_{0}(pf_{n}p) + χ_{0}(p^{⊥}f_{n}p^{⊥}). Then [q_{n}, p] = 0, n ≥ 1 and ∥q_{n} − f_{n}∥ → 0 as n → ∞. By functional calculus, there is a sequence of unitaries (v_{n}) in D such that q_{n} = v_{n}^{∗}f_{n}v_{n} and ∥v_{n} − 1∥ → 0 as n → ∞. The sequence (v_{n}) has the desired properties. ■

Lemma 5.2 Let E_{i} and F_{i} be projections in M(K(H) ⊗ B) with

∥E_{0} − E_{1}∥ < \frac{1}{2}, ∥F_{0} − F_{1}∥ < \frac{1}{2}, E_{i} − F_{i} ∈ K(H) ⊗ B

for i ∈ {0, 1}. Then [E_{0}, F_{0}] = [E_{1}, F_{1}] in KK(ℂ, B).

Proof: If X_{t} = (1 − t)E_{0} + tE_{1} and Y_{t} = (1 − t)F_{0} + tF_{1}, then sp(X_{t}), sp(Y_{t}) ⊆ [0, 1/2)∪(1/2, 1] for all t ∈ [0, 1], by Lemma 5.1(i). Define E_{t} = χ_{0}(X_{t}) and F_{t} = χ_{0}(Y_{t}). Then (E_{t}, F_{t}), 0 ≤ t ≤ 1 is a homotopy of Cuntz pairs from (E_{0}, F_{0}) to (E_{1}, F_{1}). ■

Lemma 5.3 If E, F are projections in M(K(H) ⊗ B) with E − F ∈ K(H) ⊗ B (i.e. (E, F) is a Cuntz pair), and e is a projection in K(H) ⊗ B with

∥[e, E]∥ < \frac{1}{9}  ∥[e, F]∥ < \frac{1}{9}  ∥e^{⊥}(E − F)e^{⊥}∥ < \frac{1}{9},

then the natural isomorphism between KK(ℂ, B) and K_{0}(B) takes [E, F] to

[χ_{0}(eEe)] − [χ_{0}(eF e)].

Proof: Note that ∥(eEe)^{2} − eEe∥ ≤ ∥[e, E]∥ ≤ 1/9. Therefore ∥eEe − χ_{0}(eEe)∥ ≤ λ with λ = (3 − \sqrt{5})/6 by Lemma 5.1(ii). We have similar estimates for e^{⊥}Ee^{⊥}, eF e and e^{⊥}F e^{⊥}.

Let g = χ_{0}(eEe), g′ = χ_{0}(e^{⊥}Ee^{⊥}), h = χ_{0}(eF e), h′ = χ_{0}(e^{⊥}F e^{⊥}). Then

∥E − (g + g′)∥ ≤ ∥E − eEe − e^{⊥}Ee^{⊥}∥ + ∥eEe − χ_{0}(eEe)∥ + ∥e^{⊥}Ee^{⊥} − χ_{0}(e^{⊥}Ee^{⊥}∥
≤ 2/9 + 2λ < \frac{1}{2}.

Similarly, one shows that

∥F − (h + h′)∥ ≤ 2/9 + 2λ < \frac{1}{2}, ∥g′ − h′∥ ≤ 2/9 + λ < \frac{1}{2},

and all those quantities are compact. Moreover g, g′, h, h′ are all projections and we may apply Lemma 5.2 to get

[E, F] = [g + g′, h + h′] = [g, h] + [g′, h′] = [g, h] + [g′, g′] = [g, h]

in KK(ℂ, B). Since g, h ∈ K(H) ⊗ B, the isomorphism between KK(ℂ, B) and K_{0}(B) takes [g, h] to [g] − [h]. ■

We now come to the main results in this section. The reader is referred to Section 2.2 for examples of quasidiagonal strictly nuclear nuclearly absorbing representations.

Any element α ∈ KK(A, B) induces a morphism α_{∗} : \underline{K} (A) → \underline{K} (B). Let us first describe the map induced by α on K_{0}. If A is unital and if p is a projection in M_{n}(A), and α is given by a Cuntz pair (τ, γ), then α_{∗} takes [p] ∈ K_{0}(A) ≅ KK(ℂ, A) to [(τ ⊗ id_{n})(p), (γ ⊗ id_{n})(p)] ∈ KK(ℂ, B). More generally, if D is a unital C^{∗}-algebra, then α_{D} = (τ ⊗id_{D}, γ⊗id_{D}) induces a map α_{∗} : K_{0}(A⊗D) → K_{0}(B⊗D). In particular α induces a map on \underline{K} (−).

Theorem 5.4 Let A be a unital, separable, C^{∗}-algebra and let B be a unital C^{∗}- algebra. Let α ∈ KK_{nuc}(A, B). Assume that there exists a quasidiagonal strictly nuclear unital nuclearly absorbing representation γ : A ⟶ M(K(H) ⊗ B), and let γ_{n} : A ⟶ M_{r_{n}}(B) be a quasidiagonalization of γ by (e_{n}) as in Definition 2.5.

For any \underline{K}-triple (P, F, δ) there exist N and a completely positive contraction

σ : A ⟶ M_{2r_{N}} (B)

such that σ and γ_{N} are both nuclear, δ-multiplicative on F, and satisfy

σ_{♯}(p) − (γ_{N})_{♯}(p) = α_{∗}[p]

for all p ∈ P. We may arrange that σ(1) and γ_{N}(1) are both projections.

Proof: By Lemma 3.1 we may represent α by the Cuntz pair (ρ, θ̂) consisting of strictly nuclear representations. Recall that this entails that

ρ(a) − θ̂(a) ∈ K(H) ⊗ B

for all a ∈ A.

Since γ̂ and θ̂ are both nuclearly absorbing, we have θ̂ ∼ γ̂ in the sense of Definition 2.1. Thus we get a sequence u_{i} ∈ U(M(K(H) ⊗ B)) with

γ̂(a) − u_{i}θ̂(a)u_{i}^{∗} ∈ K(H) ⊗ B  ∥γ̂(a) − u_{i}θ̂(a)u_{i}^{∗}∥ ⟶ 0. (5.1)

Note that for each i, (u_{i}ρu_{i}^{∗}, u_{i}θ̂u_{i}^{∗}) is a Cuntz pair representing α. Then X_{i} = (u_{i}ρu_{i}^{∗}, γ̂) is also a strictly nuclear Cuntz pair, and [X_{i}]_{∗} : \underline{K} (A) ⟶ \underline{K} (B) converges pointwise to the group homomorphism induced by α as a consequence of Lemma 5.2. Fix i large enough that [X_{i}]_{∗}[p] = α_{∗}[p] for each p ∈ P.

Let τ = u_{i}ρu_{i}^{∗} and let e_{n} ∈ M_{r_{n}}(B) be as in the statement. Then f_{n} = e_{n} ⊕ e_{n} ∈ M_{2r_{n}}(B) is an approximate unit of projections which in the obvious sense quasidiagonalizes γ̂ into γ_{n} : A ⟶ M_{2r_{n}}(B). We set σ_{n} = f_{n}τf_{n}; the maps thus achieved are nuclear maps by Remark 2.6(ii).

Note now that

[f_{n}, τ(a)] ⟶ 0 f_{n}^{⊥}(τ(a) − γ̂(a))f_{n}^{⊥} ⟶ 0

for a ∈ A, since τ(a) − γ̂(a) ∈ K(H) ⊗ B. We have that P ⊆ C where

C = ⨁_{m≤M} A ⊗ C(𝕋) ⊗ C(W_{m}) ⊗ M_{K}.

Therefore for N large enough, σ_{N} is δ-multiplicative on F and

∥[f_{N} ⊗ 1_{C}, (τ ⊗ id_{C})(p)]∥ < \frac{1}{9} ∥[f_{N} ⊗ 1_{C}, (γ̂ ⊗ id_{C})(p)]∥ < \frac{1}{9}
∥f_{N}^{⊥} ⊗ 1_{C}((τ ⊗ id_{C})(p) − (γ̂ ⊗ id_{C})(p))f_{N}^{⊥} ⊗ 1_{C} ∥ < \frac{1}{9},

for all p ∈ P. By Section 3.3 and Lemma 5.3 we have

(σ_{N})_{♯}(p) − (γ_{N})_{♯}(p) =
[χ_{0}(f_{N} ⊗ 1_{C}(τ ⊗ id_{C})(p)f_{N} ⊗ 1_{C})] − [χ_{0}(f_{N} ⊗ 1_{C}(γ̂ ⊗ id_{C})(p)f_{N} ⊗ 1_{C})] =
[(τ ⊗ id_{C})(p), (γ̂ ⊗ id_{C})(p)] = [X_{i}]_{∗}[p] = α_{∗}[p]

By Lemma 5.1(iii) there is a sequence of unitaries (v_{n}) in M(K(H) ⊗ B) such that [v_{n}τ(1)v_{n}^{∗}, f_{n}] = 0 and ∥v_{n} − 1∥ → 0 as n → ∞. By replacing σ_{n} = f_{n}τf_{n} by σ_{n} = f_{n}v_{n}τv_{n}^{∗}f_{n}, we have that σ_{n}(1) is a projection. Note that if N is large enough, (f_{N}τf_{N})_{♯}(p) = (f_{N}v_{N}τv_{N}^{∗}f_{N})_{♯}(p), p ∈ P, by Lemma 3.13. ■

Next we specialize the result to the case of a quasidiagonal source A. An application of the general existence result for purely infinite C^{∗}-algebras can be found in Theorem 6.17.

We say, cf. Definition 2.10, that A is locally RFD if for any finite set F and any ε > 0, there exists an RFD subalgebra A′ of A, containing all elements of F up to ε.

Theorem 5.5 Let A, B be unital C^{∗}-algebras with A quasidiagonal, and let α ∈ KK_{nuc}(A, B). For any \underline{K}-triple (P, F, δ) there exist N and completely positive nuclear contractions

σ : A ⟶ M_{N}(B) μ : A ⟶ M_{N}(ℂ1_{B})

which are δ-multiplicative on F and satisfy

σ_{♯}(p) − μ_{♯}(p) = α_{∗}[p]

for all p ∈ P. We may arrange that σ(1) and μ(1) are both projections. Moreover if A is locally RFD and if ε > 0 is given, we can arrange that there is a unital RFD subalgebra D of A such that F ⊆_{ε} D and the restriction of μ to D is a ∗-homomorphism.

Proof: Since A is quasidiagonal, it has a quasidiagonal admissible scalar representation θ. Note that, using projections e_{n} ∈ K(H) ⊗ 1_{B} we obtain a quasidiagonalization consisting of maps θ_{n} : A ⟶ M_{r_{n}}(ℂ1_{B}). The first part follows from Theorem 5.4, by taking μ = θ_{n} for some large n.

In case A is locally RFD, we find first an unital RFD subalgebra D of A such that F ⊆_{ε_{1}} D. Then we work with an admissible representation θ whose restriction to D is a direct sum of finite dimensional representations. It is then clear that one can choose the quasidiagonalization such that the restriction of θ_{n} to D is a ∗-homomorphism for all n. ■

6 Classification

In this section, we present applications of the uniqueness and existence results to classification problems. The first part is devoted to a class of finite C^{∗}-algebras which allow a further refinement of the uniqueness result, leading to a complete classification of C^{∗}-algebras in this class having K_{0}-group ℚ. In the last part, we show how the results also apply to reprove the classification theorem for purely infinite C^{∗}-algebras by Kirchberg and Phillips.

6.1 Approximate unitary equivalence

In a class of C^{∗}-algebras studied by H. Lin it is possible to absorb the stabilization required in Theorem 4.15, leading to further improved uniqueness results. Furthermore, this class is contained in our class of admissible targets. In this section, we develop these points.

Definition 6.1 ([35]) A simple unital C^{∗}-algebra is called TAF (tracially approximately finite dimensional) if for any finite subset F ⊆ A, any ε > 0, and any nonzero projection q ∈ A there is a projection p ∈ A, p ≠ 1_{A}, and there is a finite dimensional C^{∗}-algebra C ⊆ p^{⊥}Ap^{⊥} with p^{⊥} ∈ C such that

(i) ∥[p, a]∥ < ε for all a ∈ F.

(ii) dist(p^{⊥}ap^{⊥}, C) < ε for all a ∈ F.
(iii) upu^{∗} ≤ q for some unitary u ∈ A.

Example 6.2 The real rank zero approximately (sub)homogeneous C^{∗}-algebras classified in [20] and [15] are all TAF algebras. Also the class of examples of nonnuclear subalgebras of AF algebras constructed in [12] consists entirely of TAF algebras.

The conditions (i)–(ii) from [42] imply quasidiagonality of A. Let us summarize a few other structural results on TAF C^{∗}-algebras that we shall need. Lemmas 6.3 and 6.5 are from [35].

Lemma 6.3 ([35, 3]) Let A be a simple unital TAF C^{∗}-algebra. Then A has real rank zero, stable rank one, and K_{0}(A) is weakly unperforated in the sense of [4]. When p ∈ A is a projection and n ∈ ℕ, then both pAp and M_{n}(A) are simple unital TAF C^{∗}-algebras.

Lemma 6.4 Let B be an infinite dimensional unital separable simple C^{∗}-algebra of real rank zero, stable rank one, with K_{0}(B) weakly unperforated. Then for any n ≥ 1 and any nonzero projection f ∈ B there are mutually orthogonal projections e^{1}, …, e^{n} and r in B such that e^{1} + ⋯ + e^{n} + r = 1_{B} with [e^{1}] = ⋯ = [e^{n}] and [r] < [f].

Proof: Let QT (B) denote the normalized quasitraces on B. The image of the natural map ρ_{0} : K_{0}(B) ⟶ Aff(QT (B)) is uniformly dense by [4, 6.9.3] and K_{0}B has the strict ordering induced from ρ_{0} by [4, 6.9.2]. If e is a projection we write ê = ρ_{0}(e). By simplicity, we find N big enough such that N[f] > [1]. If ε = 1/nN, then 1/n − ε > 0 and nε ˆ1 < f̂. Since the image of ρ_{0} is uniformly dense, there is a projection e ∈ B such that (1/n − ε)ˆ1 < ê < 1/n ˆ1. Therefore 0 < 1 − nê < nεˆ1 < f̂, hence

0 < [1] − n[e] < [f].

This is readily seen to imply the statement. Indeed if d^{i} ∈ B are projections equivalent to e then d^{1} ⊕ ⋯ ⊕ d^{n} is equivalent to a subprojection d of 1_{B}. If r = 1 − d, then [r] = [1] − n[e] < [f]. ■

The TAF C^{∗}-algebras are prone to classification because of the following factorization property. Lemma 6.5 ([35]) Let A be a simple unital TAF C^{∗}-algebra. For any n ≥ 1, any finite subset F ⊆ A and any ε > 0, there are projections p, q ∈ A with p^{⊥}Ap^{⊥} ≅ M_{n}(qAq) and [p] ≤ [q] and such that there exists an approximate factorization of id_{A}

[figure: A; ν; ^{id^{A}}; A; pAp ⊕ M_{n}(C); μ]

with ∥μν(a) − a∥ < ε for a ∈ F, where C is a unital finite dimensional C^{∗}-subalgebra of qAq, ν(a) = pap⊕(η(a)⊗1_{n}) is ε-multiplicative on F with η : A ⟶ C a unital completely positive contractive map, and μ is a unital ∗-monomorphism whose restriction to pAp is the natural inclusion. Proof: We include a proof of this result which is somewhat different from the original proof of Lin. We don’t require A to be nuclear, but may assume that A is infinite dimensional. It also suffices to prove the statement with 8ε instead of ε. Since A is TAF we find a projection P ∈ A and a finite dimensional C^{∗}-algebra C with P^{⊥} ∈ C ⊂ P^{⊥}AP^{⊥} such that for all a ∈ F we have (i) ∥[P, a]∥ < ε (ii) P^{⊥}aP^{⊥} ∈_{ε} C (iii) (n + 2)[P] ≤ [1].

The idea of the proof is to find a unital embedding of M_{n}(ℂ) ⊕ ℂ into the relative commutant of C in P^{⊥}AP^{⊥} such that the image of ℂ is supported by a very small projection. Write C ≅ M_{m(1)} ⊕ ... ⊕ M_{m(k)} and let e_{1}, ..., e_{k} be the minimal central projections of C. Let B_{i} be the relative commutant of e_{i}Ce_{i} ≅ M_{m(i)} in e_{i}Ae_{i}. Then e_{i}Ae_{i} ≅ M_{m(i)}(B_{i}) and hence B_{i} is TAF being isomorphic to a corner of A. Let f be a nonzero projection in A with (n + 1)k[f] < [P]. For each 1 ≤ i ≤ k we apply Lemma 6.4 for B_{i}. We obtain:

e_{i} = e_{i}^{1} + ⋯ + e_{i}^{n} + r_{i}

where e_{i}^{j}, 1 ≤ j ≤ n are mutually equivalent projections in B_{i} and [r_{i}] < [f] in K_{0}(A). Set e^{j} = e_{1}^{j} + ⋯ + e_{k}^{j}, 1 ≤ j ≤ n, r = r_{1} + ⋯ + r_{k} and e = e^{1} + ⋯ + e^{n}. Note that P^{⊥} = e^{1} + ⋯ + e^{n} + r with e^{j} mutually equivalent in the relative commutant of C in P^{⊥}AP^{⊥}. We have (n + 1)[r] = (n + 1)([r_{1}] + ⋯ + [r_{k}]) < (n + 1)k[f] < [P]. Therefore

(n + 1)([P] + [r]) < (n + 2)[P] ≤ [1] = [P] + n[e^{1}] + [r]

hence n([P] + [r]) < n[e^{1}]. By weak unperforation we get [P + r] ≤ [e^{1}]. We are now ready to complete the proof. By Arveson’s extension theorem [2], the inclusion C ↪ A extends to a completely positive contraction E : A → C. We have E(x) = x for x ∈ C hence ∥a − E(a)∥ ≤ 2 dist(a, C) for a ∈ A. Using (i) and (ii) we have for a ∈ F

a ∼_{2ε} P aP + P^{⊥}aP^{⊥} ∼_{2ε} P aP + E(P^{⊥}aP^{⊥})
= P aP + E(P^{⊥}aP^{⊥})r + E(P^{⊥}aP^{⊥})e ∼_{4ε} (P + r)a(P + r) + E(P^{⊥}aP^{⊥})e.

The last estimate follows by compressing the estimate a ∼_{4ε} P aP + E(P^{⊥}aP^{⊥}) by P + r. It follows that a ∼_{8ε} (P + r)a(P + r) + E(P^{⊥}aP^{⊥})e. We finish the proof by setting p = P + r, q = e^{1} and noting that E(P^{⊥}aP^{⊥})e is of the form η(a) ⊗ 1_{n} since e = e^{1}+⋯+e^{n} with e^{j} mutually equivalent in the relative commutant of C in P^{⊥}AP^{⊥}. ■

The uniqueness Theorems 4.12 and 4.15 apply to TAF algebras because of the following Proposition.

Proposition 6.6 A simple unital infinite-dimensional TAF C^{∗}-algebra is an admissible target algebra (of finite type).

Proof: Let B be a simple unital TAF C^{∗}-algebra. We get (i) and (ii) of Definition 4.9 by two results of Rieffel ([43], [44]). For (iii.1), note that if nx ≥ 0 then nx+n[1_{B}] > 0 and x + [1_{B}] > 0 by weak unperforation. Finally, to prove (iv.1), assume that dim(B) = ∞ and let x and n be given. The image of the natural map ρ_{0} : K_{0}(B) ⟶ Aff(QT (B)) is uniformly dense by [4, 6.9.3], so we can find z ∈ K_{0}(B) with ρ_{0}(x) − 1 < ρ_{0}(nz) < ρ_{0}(x) + 1. By [4, 6.9.2] we have x − [1_{B}] ≤ nz ≤ x + [1_{B}] so y = x − nz will work. ■

Theorem 6.7 Let A be a simple unital, exact, separable TAF C^{∗}-algebra satisfying the UCT. Then for any finite subset F ⊆ A and any ε > 0, there exists a \underline{K}-triple (P, G, δ) with the following property. For any unital simple infinite-dimensional TAF C^{∗}-algebra B, and any two unital nuclear completely positive contractions φ, ψ : A ⟶ B which are δ-multiplicative on G, with φ_{♯}(p) = ψ_{♯}(p) for all p ∈ P, there exists a unitary u ∈ U(B) such that

∥uφ(a)u^{∗} − ψ(a)∥ < ε

for all a ∈ F.

Proof: Let us begin by outlining the proof. We first construct such a unitary in the special case where φ, ψ are nuclear ∗-homomorphisms agreeing on all of \underline{K} (A). This involves invoking Lin’s factorization result 6.5 to pass to another pair of ∗- homomorphisms \overline{φ}, \overline{ψ} which are of a special form. Because Lin’s result only gives an approximate factorization, even though we start out with ∗-homomorphisms, our proof will take us to a setting where our uniqueness theorem for completely positive contractions Theorem 4.15 is needed. The general case will follow in the same way that Theorem 4.12 implies Theorem 4.15, by letting n = 0 and T = 0 in the proof of Theorem 4.15. We include a sketch for the benefit of the suspicious reader.

Part 1: Given F and ε, we are going to prove that whenever

(i) B = ∏ B_{i}/ ∑ B_{i} with each B_{i} a unital simple infinite-dimensional TAF C^{∗}- algebra (or B itself a unital simple infinite-dimensional TAF C^{∗}-algebra).

(ii) φ, ψ : A ⟶ B are unital nuclear ∗-homomorphisms (iii) φ_{∗} = ψ_{∗} : \underline{K} (A) ⟶ \underline{K} (B)

then there exists u ∈ U(B) with ∥uφ(a)u^{∗} − ψ(a)∥ < ε for all a ∈ F.

Let us thus fix n, P, G and δ by applying Theorem 4.15 to F and ε/3. Furthermore, let p, q, C, ν and μ be given by Lemma 6.5 such that ν is δ-multiplicative on G and ∥μν(a) − a∥ < ε/3 for all a ∈ F.

Step 1a: Since B has stable rank one by Lemma 6.3, it has cancellation of projections, and because φ_{∗} = ψ_{∗} we may assume, after conjugating ψ by a unitary in B, that the restrictions of φ and ψ to the finite dimensional algebra μ(ℂp ⊕ M_{n}(ℂ1_{C})) are equal. Applying μ to the matrix units of M_{n}(ℂ1_{C}) we can define matrix units (q_{ij}) in A, where q = q_{11} = 1_{C}. Let

e = φ(p) = ψ(p) f_{ij} = φ(q_{ij}) = ψ(q_{ij}), (6.1)

abbreviating f = f_{11}. Invoking cancellation again, since [p] ≤ [q], we can find a projection e_{0} and a unitary v in B with v(e + e_{0})v^{∗} = f. Let g = e_{0} ⊕ 1_{B} ∈ M_{2}(B) and note that [g] = (n + 1)·[f]. Hence an isomorphism γ : gM_{2}(B)g ⟶ M_{n+1}(fBf) can be found. Denoting the matrix units of M_{n+1}(ℂf) by f̃_{ij} with 0 ≤ i, j ≤ n we may choose γ such that

γ ([\begin{matrix}e_{0}b_{0}e_{0} \\ & ebe \end{matrix}]) = f̃_{00}v(ebe + e_{0}b_{0}e_{0})v^{∗}f̃_{00} γ ([\begin{matrix}0 \\ & f_{ij} \end{matrix}]) = f̃_{ij} (6.2)

Combining all of this, we get a ∗-homomorphism \overline{φ} fitting in a diagram

[figure: A; ^{id^{A}}; ν; A; φ; B; ι_{2}; gM_{2}(B)g; γ; pAp ⊕ M_{n}(C); μ; \overline{φ}; M_{n+1}(fBf),]

where ι_{2} sends B into the (2, 2) corner of gM_{2}(B)g. Identifying \overline{φ} using (6.1) and (6.2) we get that for all d ∈ pAp, x ∈ M_{n}(C),

\overline{φ}(d ⊕ x) = φ′′(d) ⊕ (φ′ ⊗ id_{n})(x)

where

φ′ : C ⟶ fBf φ′′ : pAp ⟶ fBf

are defined as corestrictions of φ and Ad(v)φ, respectively. Furthermore, by symmetry of (6.1), the same procedure shows that \overline{ψ} = γι_{2}ψμ : pAp ⊕ M_{n}(C) ⟶ M_{n+1}(fBf) has the form

\overline{ψ}(d ⊕ x) = ψ′′(d) ⊕ (ψ′ ⊗ id_{n})(x)

for all d ∈ pAp, x ∈ M_{n}(C).

Step 1b: With ι_{f} : fBf ⟶ B we clearly have that (ι_{f}φ′)_{∗} = (ι_{f}ψ′)_{∗} and (ι_{f}φ′′)_{∗} = (ι_{f}ψ′′)_{∗}. But since f is full in B as the image of a full projection under a unital map, we get by [7] that ι_{f} induces an isomorphism from \underline{K} (fBf) to \underline{K} (B), so that (φ′)_{∗} = (ψ′)_{∗} and (φ′′)_{∗} = (ψ′′)_{∗}. From this we may assume, after conjugating ψ by a unitary of fBf, that φ′ and ψ′ agree on C. Thus the maps \overline{φ}ν and \overline{ψ}ν are of the form

\overline{φ}ν(a) = [ \begin{matrix}φ′′ω(a) \\ & φ′η(a) \\ & & . & . & . \\ & & & & & φ′η(a) \end{matrix} ] \overline{ψ}ν(a) = [ \begin{matrix}ψ′′ω(a) \\ & φ′η(a) \\ & & . & . & . \\ & & & & & φ′η(a) \end{matrix} ]

where we define ω : A ⟶ pAp by ω(a) = pap. Note that ω is δ-multiplicative on G. Now φ′′ω, ψ′′ω and φ′η are δ-multiplicative on G and by Lemma 3.10 we have that (φ′′ω)_{♯}(p) = (ψ′′ω)_{♯}(p) for all p ∈ P. Therefore, Theorem 4.15 applies to the triple of maps (φ′′ω, ψ′′ω, φ′η) if we can prove that fBf is an admissible target. When B itself is a TAF C^{∗}-algebra, so is fBf by Lemma 6.3, and Proposition 6.6 applies. When B = ∏ B_{i}/ ∑ B_{i} we note that there are projections f_{i} ∈ B_{i} such that fBf is isomorphic to

∏ f_{i}B_{i}f_{i}/ ∑ f_{i}B_{i}f_{i}

and hence it is admissible of finite type by Lemma 3.4 since all the f_{i}B_{i}f_{i} are. Thus, by Theorem 4.15, there is a partial isometry v ∈ M_{n+1}(fBf) such that v^{∗}v = \overline{φ}ν(1), vv^{∗} = \overline{ψ}ν(1) and ∥v\overline{φ}ν(a)v^{∗} − \overline{ψ}ν(a)∥ < ε/3 for all a ∈ F. Since \overline{φ}ν(1) = \overline{ψ}ν(1) = γι_{2}(1) and γι_{2}(B) = γι_{2}(1)M_{n+1}(fBf)γι_{2}(1), we have that v = γι_{2}(u) for some unitary u ∈ B. Since γι_{2} is isometric we obtain ∥uφμν(a)u^{∗} − ψμν(a)∥ < ε/3, hence ∥uφ(a)u^{∗} − ψ(a)∥ < ε for all a ∈ F, since ∥μν(a) − a∥ < ε/3 for all a ∈ F.

Part 2: As the argument reducing to the case covered in Part 1 closely parallels that in the proof of Theorem 4.15, we only sketch it here. If the theorem is false, we can choose sequences P_{i}, δ_{i}, G_{i} with the properties (i)–(iii) of that proof, and corresponding simple unital TAF C^{∗}-algebras B_{i} as well as unital completely positive contractions φ_{i}, ψ_{i} : A ⟶ B_{i} being δ_{i}-multiplicative on G_{i}, satisfying (φ_{i})_{♯}(p) = (ψ_{i})_{♯}(p) for all p ∈ P_{i}; yet

inf_{u∈U(B)} max_{a∈F} ∥uφ_{i}(a)u^{∗} − ψ_{i}(a)∥ ≥ ε.

Define Φ̇, Ψ̇ : A ⟶ ∏ B_{i}/ ∑ B_{i} and check as in the proof of Theorem 4.15 that Φ̇ and Ψ̇ are ∗-homomorphisms inducing the same map on \underline{K} (A). Since (i)–(iii) of Part 1 are met, we conclude from the first part of the proof that there is a unitary U ∈ U(∏ B_{i}/ ∑ B_{i}) such that ∥UΦ̇(a)U^{∗} − Ψ̇(a)∥ < ε for all a ∈ F. Lifting U to a unitary (u_{i}) ∈ ∏ B_{i} and projecting, we get the desired contradiction. ■

Notes 6.8 As noted above, Definition 6.1 is due to H. Lin. It was motivated by a result of Popa [42] and the classification theory of AH algebras ([20], [15]). Definition 6.1 is in fact a version of the original definition, slightly simplified for the class of simple algebras. To correlate this with [35] one compares to [35, 3.7] and notes that since Popa’s conditions imply the (SP) property (all hereditary subalgebras have a nonzero projection) it suffices to find a p inside a generic corner rather than inside a generic hereditary subalgebra.

6.2 Classification results

We begin this section by presenting a shape type isomorphism result for simple TAF C^{∗}-algebras. Note that in this setting, there is no need to appeal to our existence results, as the KK-classes are represented by completely positive contractions from the outset. We then prove, this time using both existence and uniqueness, that certain TAF C^{∗}-algebras are isomorphic to the AD algebra with the same K-theory. This leads to our main classification result, Theorem 6.13. We refer the reader to [8] for the basics of asymptotic morphisms.

Theorem 6.9 Let A and B be two unital, separable, nuclear, simple TAF C^{∗}-algebras satisfying the UCT. Suppose that there are unital asymptotic morphisms φ = (φ_{t}) : A ⟶ B and ψ = (ψ_{t}) : B ⟶ A such that φ_{∗} : \underline{K} (A) ⟶ \underline{K} (B) is bijective and φ∗^{−1} = ψ_{∗}. For instance this condition is satisfied if A is homotopy equivalent, or just shape equivalent, to B. Then A is isomorphic to B.

Proof: We may assume that both A and B are infinite-dimensional. We use repeatedly Theorem 6.7 to find an increasing sequence of positive numbers t_{n} ∈ (0, 1) and sequences of unitaries u_{2n−1} ∈ A, u_{2n} ∈ B, n ≥ 1 such that the following diagram is a two-sided approximate intertwining in the sense of Elliott [23].

\xymatrix{A \ar[rr]^{Ad(u_{1})} \ar[dr]_{φ_{t_{1}}} &  & A \ar[rr]^{Ad(u_{3})} \ar[dr]_{φ_{t_{3}}} &  & A \ar[dr] \ar[r] & ... \\  & B \ar[rr]_{Ad(u_{2})} \ar@{-->}[ur]^{ψ_{t_{2}}} &  & B \ar@{-->}[ur]^{ψ_{t_{4}}} \ar[rr]_{Ad(u_{4})} &  & B \ar[r] & ...}

■

If r is a positive integer we denote by 𝕀_{r}^{∼} the C^{∗}-algebra of continuous functions f : [0, 1] ⟶ M_{r}(ℂ) such that f(0), f(1) ∈ ℂ1_{r}. We denote by 𝕀_{r} the subalgebra of functions vanishing at 0. Let D be the class of algebras B of the form B = B_{1}⊕⋯⊕B_{n} where each B_{i} is either a circle algebra M_{k}(C(S^{1})) or a dimension-drop algebra M_{k}(𝕀_{r}^{∼}). An AD algebra is a C^{∗}-algebra which is isomorphic to an inductive limit of a sequence of C^{∗}-algebras in D.

Lemma 6.10 Let D be a dimension drop algebra D = 𝕀_{n} or D = C_{0}(ℝ). Let E be a simple unital TAF algebra and let f be a nonzero projection in E. Then the map [D, fEf] ⟶ KK(D, E) is surjective.

Proof: Let n = 1 if D = C_{0}(ℝ). The proof uses the following series of facts:

(i) ([19]) lim_{⟶}[D, M_{k}(E)] = KK(D, E).

(ii) ([17] when D = 𝕀_{n}) If C is a finite dimensional C*-algebra, and η : D ⟶ C is a ∗-homomorphism, then the map d ↦ η(d) ⊗ 1_{n} from D ⟶ M_{n}(C) is null homotopic.

(iii) ([37]) There is a finite subset F ⊆ D and there is ε > 0 such that if α, β : D ⟶ B are two ∗-homomorphisms satisfying ∥α(d) − β(d)∥ < ε, then α is homotopic to β.

(iv) ([37]) For any finite subset F ⊆ D and ε > 0 there is a finite subset F_{1} ⊆ D and there is ε_{1} > 0 such that if α : D ⟶ B is any completely positive contractive map which is ε_{1}-multiplicative on F_{1}, then there exists a ∗-homomorphism β : D ⟶ B with ∥α(d) − β(d)∥ < ε/2 for all d ∈ F.

Fix F, ε as in (iii) and let F_{1}, ε_{1} be given by (iv). We may assume that F ⊆ F_{1} and ε > ε_{1}. Let x ∈ KK(D, E). Then by (i) x can be represented by some ∗-homomorphism γ : D ⟶ M_{k}(E). With A = M_{k}(E), A is a simple unital TAF C^{∗}-algebra by Lemma 6.3. Since E is simple there is m ≥ 1 such that [1_{E}] ≤ m[f]. Consider an approximation of id_{A} provided by Lemma 6.5 applied for the set γ(F_{1}), ε_{1}/2, and the integer mnk:

[figure: A; ν; ^{id^{A}}; A; pAp ⊕ M_{mnk}(C); μ]

Therefore

∥μν(a) − a∥ < ε_{1}/2 (6.3)

for a ∈ γ(F_{1}), C a unital finite dimensional C^{∗}-subalgebra of qAq, ν(a) = ω(a) ⊕ (η(a) ⊗ 1_{mnk}), ω(a) = pap, ν is ε_{1}-multiplicative on γ(F_{1}) and η : A ⟶ C is a unital completely positive contractive map. We may arrange that μ(p) ≤ f. Indeed, from [p] ≤ [q] and [p] + mnk[q] = 1 we see that (mnk + 1)[p] ≤ [1]. Since μ(1) = 1_{A}, we obtain (mnk + 1)[μ(p)] ≤ [1_{A}] = n[1_{E}] ≤ mnk[f], hence [μ(p)] ≤ n[μ(p)] < [f] since K_{0}(E) is weakly unperforated in the sense of [4] by Lemma 6.3. After conjugating μ by a suitable unitary, we obtain μ(p) ≤ f.

Let us observe that ωγ and ηγ are ε_{1}-multiplicative on F_{1}. By (iv) there are ∗- homomorphisms ω′ : D ⟶ pAp and η′ : D ⟶ C such that if we set ν′ = ω′ ⊕ (η′ ⊗ 1_{mn}), then

∥νγ(d) − ν′(d)∥ < ε/2 (6.4)

for all d ∈ F. From (6.3) we have ∥μνγ(d) − γ(d)∥ < ε_{1}/2 for d ∈ F_{1}. Combining this with (6.4) we get

∥μν′(d) − γ(d)∥ < (ε_{1} + ε)/2 < ε

for all d ∈ F_{1} ∩ F = F. By (iii) this implies that γ is homotopic to μν′. By (ii) ν′ is homotopic to ω′ so that γ is homotopic to μω′. We conclude the proof by observing that the image of μω′ is contained in fEf since μω′(1) = μ(p) ≤ f. ■

Lemma 6.11 Let A be a unital C^{∗}-algebra with (K_{0}(A), [1_{A}]) = (ℚ, 1). Any finite set of projections P_{0} ⊆ A ⊗ K can be complemented to a K_{0}-triple (P_{0}, G, δ) with the property that for any unital completely positive contraction φ : A → A which is δ-multiplicative on G, one has φ_{♯}(p) = [p] for all p ∈ P_{0}.

Proof: We may write [p] = \frac{r}{s}[1_{A}], so s·p ⊕ m·1_{A} ∼ (r + m)·1_{A} for some m ≥ 0. When φ is sufficiently multiplicative, we have

sφ_{♯}(p) + m[1_{A}] = φ_{♯}(s·p ⊕ m·1_{A}) = φ_{♯}((r + m)·1_{A}) = (r + m)[1_{A}].
■

The following theorem generalizes a result of Lin [35] by the fact that it allows non-zero (countable) K_{1}-groups.

Theorem 6.12 Let A be a unital, separable, nuclear and simple TAF C^{∗}-algebra satisfying the UCT and suppose that K_{0}(A) ≅ ℚ as ordered groups. Then A is isomorphic to an AD-algebra.

Proof: Clearly A is infinite-dimensional, and we may assume that (K_{0}(A), [1_{A}]) ≅ (ℚ, 1) as ordered pointed groups. For any finite subset F ⊆ A and any ε > 0 we will find an algebra B ∈ D, and a ∗-homomorphism β : B ⟶ A such that F ⊆_{ε} β(B). This will prove the theorem as all elements of the class D (introduced before Lemma 6.10) are semiprojective [37]. As noted in Remark 4.16, applying Theorem 4.15 to fixed F and ε associated to C^{∗}-algebras A, B with torsion-free divisible K_{0}-groups results in a K_{∗}-triple (P, G, δ) rather than a general \underline{K}-triple.

Let (P_{0}, G_{0}, δ_{0}) and (V, G_{1}, δ_{1}) be a K_{0}-triple and a K_{1}-triple, respectively, given by Lemma 3.11 for the input K_{∗}-triple (P, G, δ). Let (P′, G′, δ′) be a K_{∗}-triple given by Lemma 3.12 for the input triples (P, G_{0}, δ_{0}) and (V, G_{1}, δ_{1}). We also may assume that G′ and δ′ satisfy the conclusion of Lemma 6.11 applied for the unital C^{∗}-algebra A and the set of projections P_{0}.

By [23] there is an AD algebra D and a group isomorphism κ : K_{1}(A) ⟶ K_{1}(D). By Theorem 5.5 there exist completely positive contractive maps σ : A ⟶ M_{N}(D) and μ : A ⟶ M_{N}(ℂ1_{D}) which are δ′-multiplicative on G′ and σ_{♯}(p′) − μ_{♯}(p′) = κ[p′] for all p′ ∈ G′. Here κ is regarded as an element of Hom(K_{∗}(A), K_{∗}(B)) ≅ Hom_{Λ}(\underline{K} (A) , \underline{K} (B)). By the choice of the K_{∗}-triple (P′, G′, δ′) we have that σ_{♯}(u) − μ_{♯}(u) = κ[u] for all unitaries u ∈ V. Note that μ_{♯}(u) = 0 since K_{1}(ℂ) = 0, so that we have σ_{♯}(u) = κ[u] for all u ∈ V.

Recall that σ(1_{A}) = q is a projection, so that if we set B = qM_{N}(D)q, then σ : A ⟶ B is a unital map. Write B as the inductive limit of an increasing sequence of algebras B_{k} ∈ D, and let j_{k} : B_{k} ⟶ B be the inclusion map. Using the Choi-Effros theorem as in Lemma 4.2 of [16] we find a sequence of completely positive contractive maps η_{k} : B ⟶ B_{k} such that j_{k}η_{k} converges to id_{B} in the point-norm topology. Choose k large enough so that

σ_{♯}(u) = (j_{k}η_{k}σ)_{♯}(u) (6.5)

for all u ∈ V.

Consider the group morphism κ^{−1}(j_{k})_{∗} : K_{1}(B_{k}) ⟶ K_{1}(A). There is a a unital ∗-homomorphism γ : B_{k} ⟶ A such that γ_{∗} = κ^{−1}(j_{k})_{∗}, obtained as follows. Write B_{k} = M_{ℓ(1)}(B′_{k1}) ⊕ ⋯ ⊕ M_{ℓ(r)}(B′_{kr}) where the B′_{ki} are either C(S^{1}) or 𝕀_{n(i)}^{∼}. Let q_{1} + ⋯ + q_{r} = 1_{A} be a partition of 1_{A} by nonzero projections. We have K_{0}(q_{i}Aq_{i}) ≅ K_{0}(A) ≅ ℚ as ordered groups. Therefore we find mutually orthogonal projections p_{i} such that ℓ(i)[p_{i}] ≅ [q_{i}]. Since A has cancellation of projections we find a unital inclusion

⨁_{i=1}^{r} M_{ℓ(i)}(p_{i}Ap_{i}) ⊆ A.

Using Lemma 6.10 we find unital ∗-homomorphisms γ_{i} : B_{i} ⟶ p_{i}Ap_{i} such that γ = ⊕_{i=1}^{r}(γ_{i} ⊗ id_{ℓ(i)}) has the desired property.

Next we want to show that γ ∘ (η_{k}σ) gives an approximate factorization of id_{A} on K_{∗}(A). More precisely we want that (γη_{k}σ)_{♯}(p) = [p] for all p ∈ P. By virtue of our choice of the K_{0} and K_{1}-triples above, it suffices to show that (γη_{k}σ)_{♯}(p_{0}) = [p_{0}] for all p_{0} ∈ P_{0} and (γη_{k}σ)_{♯}(u) = [u] for all u ∈ V. Using Lemma 3.10 twice, the definition of γ and (6.5) we have

(γη_{k}σ)_{♯}(u) = γ_{∗}(η_{k}σ)_{♯}(u) = κ^{−1}(j_{k})_{∗}(η_{k}σ)_{♯}(u) = κ^{−1}(j_{k}η_{k}σ)_{♯}(u) = κ^{−1}σ_{♯}(u) = [u]

for all u ∈ V. It remains to check that (γη_{k}σ)_{♯}(p_{0}) = [p_{0}] for all p_{0} ∈ P_{0}, but this follows from Lemma 6.11 by our choice of the G′ and δ′.

Define α = η_{k}σ : A ⟶ B_{k}. We have seen that (γα)_{♯}(p) = [p] for all p ∈ P. Therefore by Theorem 4.15 there is unitary u ∈ A such that if β = Ad(u) ∘ γ, then ∥βα(a) − a∥ < ε for all a ∈ F, hence F ⊆_{ε} β(B). ■

Theorem 6.13 Let A, B be unital, separable, nuclear and simple TAF C^{∗}-algebras satisfying the UCT. Suppose that (K_{0}(A), [1_{A}]) ≅ (K_{0}(B), [1_{B}]) ≅ (ℚ, 1) as ordered groups, and K_{1}(A) ≅ K_{1}(B). Then A is isomorphic to B.

Proof: By Theorem 6.12 both A and B are isomorphic to simple real rank zero AD algebras. These are classified by their K-theory data as proved by Elliott [23]. ■

Notes 6.14 Lin proved Theorem 6.12 for K_{1}(A) = 0 in [35]. He subsequently, independently from and at about the same time as the present work, generalized his result to allow general K_{1}-groups in [36]. This paper also attempts to give classification of C^{∗}-algebras with other K_{0}-groups.

6.3 Purely infinite C^{∗}-algebras

The purpose of this section is to demonstrate how our methods can be applied to give the classification result of Kirchberg and Phillips starting from three basic, albeit deep, structural results about purely infinite C^{∗}-algebras. The methods used here are very similar to those used in the finite case, with Cuntz’ algebra O_{2} playing the role of M_{n}(ℂ). The exposition will emphasize this similarity. To make it very clear exactly how much we need to import from the theory of this class of C^{∗}-algebras we collect the required results below.

(I) ([31, 2.8]) Any exact, separable and unital C^{∗}-algebra embeds unitally into O_{2}.

(II) ([45, 3.6]) Let B be an admissible target algebra. If φ, ψ : O_{2} ⟶ B is a pair of unital ∗-homomorphisms, then for any finite set F ⊆ O_{2} and any ε > 0 there exists u ∈ U(B) with ∥uφ(a)u^{∗} − ψ(a)∥ < ε for all a ∈ F.

(III) (A variation of [41, 2.4]) Let A be a purely infinite nuclear separable unital C^{∗}- algebra. For any n ≥ 1, any finite subset F ⊆ A and any ε > 0, there is a projection p ∈ A with p^{⊥}Ap^{⊥} ≅ M_{n}(O_{2}) such that there exists an approximate factorization of id_{A}

[figure: A; ν; ^{id^{A}}; A; pAp ⊕ M_{n}(O_{2}); μ]

with ∥μν(a) − a∥ < ε for a ∈ F, ν(a) = pap ⊕ (η(a) ⊗ 1_{n}) is ε-multiplicative on F with η : A ⟶ O_{2} a unital completely positive contractive map, and μ is a unital ∗-monomorphism whose restriction to pAp is the natural inclusion.

Remark 6.15 We have rephrased (II) and (III) slightly to suit our needs. Rørdam requires that B satisfies

U(B)/U_{0}(B) ≅ K_{1}(B)

and has finite exponential lenght ([40]), but this follows by Definition 4.9 following our observations in Section 3.2. Also, Phillips proves (III) only for n = 1, but if we write μ : O_{2} ⟶ M_{n}(O_{2}) for the map sending x to Diag(x, …, x) and choose a unital embedding ι : M_{n}(O_{2}) ⟶ O_{2}, then (II) shows that a unitary u ∈ U(O_{2}) exists making the diagram

\xymatrix{O_{2} \ar[dr]_{μ} \ar[rr]^{Ad(u)} &  & O_{2} \\  & M_{n}(O_{2}) \ar@{-->}[ur]_{ι}}

commute up to ε on F. Thus we may replace O_{2} by M_{n}(O_{2}) without loss of generality (and without using the theorem we are aiming for).

Apart from (I)–(III), all we need to know about a purely infinite C^{∗}-algebra A is that it has real rank zero ([52]), that the canonical maps from Proj(A) and U(A) to K_{0}(A) and K_{1}(A) are surjective ([10]), and furthermore, that if p, q ∈ A are nonzero projections, then

[p] = [q] ⟹ p ∼ q.

The latter fact is also from [10]. Note that this shows:

Proposition 6.16 A purely infinite simple unital C^{∗}-algebra is an admissible target (of infinite type).

Theorem 6.17 Let A, B be unital C^{∗}-algebras with A exact and separable and B containing a unital copy of O_{2}, and let α ∈ KK_{nuc}(A, B). Then for any \underline{K}-triple (P, F, δ) there exists a nuclear completely positive contraction σ : A ⟶ B which is δ-multiplicative on F and satisfies σ_{♯}(p) = α_{∗}[p] for all p ∈ P. We may arrange that σ(1) is a projection.

Proof: By (I), A embeds unitally into O_{2}, so with ι defined as the composite

\xymatrix{A \ar@{^{(}->}[r] & O_{2} \ar@{^{(}->}[r] & B}

we obtain a unital full embedding. By Theorem 2.22, the representation d_{ι} is absorbing, and it is clearly also quasidiagonal as it commutes with the projections e_{n} = n·1_{B}. By Theorem 5.4, we find N and σ : A ⟶ M_{2N}(B) such that

(σ)_{♯}(p) − (γ_{N})_{♯}(p) = α_{∗}[p]

where γ_{N} is a ∗-homomorphism of the form

γ_{N}(a) = 0 ⊕ N ·ι(a),

so that (γ_{N})_{∗} = 0 by the fact that ι factors through O_{2}. Thus (σ)_{♯} = α_{∗} on P, and assuming, as we may, that σ(1) is a projection e, we have that e ∈ M_{2N+1}(B) is a proper subprojection of (2N + 1)·1_{B} which is equivalent to a subprojection of 1_{B} = 1_{O_{2}} via some unitary u. We may hence replace σ by uσu^{∗} : A ⟶ B inducing the same partial map on P. ■

Remark 6.18 If B is simple purely infinite and α_{∗}[1] = [1], then after taking 1 ∈ P we may arrange that σ is unital. Indeed, since σ(1) = e is a projection and σ_{♯}[1] = α_{∗}[1] = [1], [1 − e]=0, hence (1 − e)B(1 − e) contains a unital copy of O_{2}. Let ȷ : A → (1 − e)B(1 − e) be a unital embedding with image contained in this copy of O_{2}. Replacing σ by σ ⊕ ȷ we have that σ_{♯} = (σ ⊕ ȷ)_{♯} on P and σ ⊕ ȷ is unital.

Remark 6.19 Kirchberg proved a stronger form of the previous theorem where α is lifted to a ∗-monomorphism.

Theorem 6.20 Let A be a purely infinite separable unital exact C^{∗}-algebra satisfying the UCT. Then for any finite subset F ⊆ A and any ε > 0, there exists a \underline{K}-triple (P, G, δ) with the following property. For any purely infinite simple unital C^{∗}-algebra B, and any two unital nuclear completely positive contractions φ, ψ : A ⟶ B which are δ-multiplicative on G, with φ_{♯}(p) = ψ_{♯}(p) for all p ∈ P, there exists a unitary u ∈ U(B) such that ∥uφ(a)u^{∗} − ψ(a)∥ < ε for all a ∈ F.

Proof: The proof follows closely that of Theorem 6.7, and we are only going to indicate the changes needed. Here Part 1 of the proof deals with a pair of ∗-homomorphisms into B = ∏ B_{i}/ ∑ B_{i} with each B_{i} a purely infinite C^{∗}-algebra. Applying first Theorem 4.15 and then (III) we get n, P, G, δ, p, q, ν and μ as in that proof. Then Step 1a applies verbatim as soon as one notes that because they are all nonzero, one has cancellation on all the projections in play by the result of Cuntz mentioned above. In Step 1b one applies (II) to obtain a unitary of fBf which conjugates ψ′ to φ′ to within ε/3 on F. This suffices to achieve the desired conclusion by the argument given in Theorem 4.15. Furthermore, to get that fBf is admissible of infinite type, one uses the same argument to reduce to the case of showing that a corner of a purely infinite C^{∗}-algebra is also admissible of infinite type. This is clear since it is itself purely infinite. Finally, Part 2 of the proof carries through verbatim because of Proposition 6.16. ■

Theorem 6.21 ([30], [39]) Let A and B be purely infinite separable unital nuclear C^{∗}- algebras satisfying the UCT. Then any isomorphism κ : (K_{∗}(A), [1_{A}]) ⟶ (K_{∗}(B), [1_{B}]) is induced by a ∗-isomorphism.

Proof: We may assume that A and B are in the standard form, that is 1_{A} and 1_{B} both represent the zero class in their respective K_{0}-groups. It follows by [10] that A and B both contain unital copies of O_{2}. We may thus apply the existence result Theorem 6.17 in conjunction with Remark 6.18 to get unital maps σ_{i} : A ⟶ B and τ_{i} : B ⟶ A which are increasingly multiplicative on larger and larger sets, and induce κ and κ^{−1}, respectively, on larger and larger subsets of \underline{K} (A) and \underline{K} (B). Arranging this appropriately, we may conclude by the uniqueness result Theorem 6.20 that unitaries u_{i}, v_{i} exist making

\xymatrix{A \ar[rr]^{Ad(u_{1})} \ar[dr]_{σ_{1}} &  & A \ar[rr]^{Ad(u_{2})} \ar[dr]_{σ_{2}} &  & A \ar[dr] \ar[r] & … \\  & B \ar[rr]_{Ad(v_{1})} \ar@{-->}[ur]^{τ_{1}} &  & B \ar@{-->}[ur]^{τ_{2}} \ar[rr]_{Ad(v_{2})} &  & B \ar[r] & …}

an approximate intertwining in the sense of Elliott ([23]). ■

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Marius Dadarlat Department of Mathematics Purdue University West Lafayette IN 47907 U.S.A.

         mdd@math.purdue.edu

Søren Eilers Matematisk Afdeling Københavns Universitet Universitetsparken 5 DK-2100 Copenhagen Ø Denmark

                                           eilers@math.ku.dk