Approximate unitary equivalence and the topology of Ext(A, B)

Marius Dadarlat ^{∗}

Department of Mathematics, Purdue University, West Lafayette IN 47907

October 5, 1999

Abstract

Let A, B be unital C*-algebras and assume that A is separable and quasidiagonal relative to B. Let φ, ψ : A → B be unital ∗-homomorphisms. If A is nuclear and satisfies the UCT, we prove that φ is approximately stably unitarily equivalent to ψ if and only if φ_{∗} = ψ_{∗} : K_{∗}(A, ℤ/n) → K_{∗}(B, ℤ/n) for all n ≥ 0. We give a new proof of a result of [DE_{2}] which states that if A is separable and quasidiagonal relative to B and if φ, ψ : A → B have the same KK-class, then φ is approximately stably unitarily equivalent to ψ. For nuclear separable C*-algebras A, we give a KKtheoretical description of the closure of zero in Ext(A, B).

1 Introduction

Two representations γ, γ′ : A → 𝕄(K ⊗ B) are called properly approximately unitarily equivalent, written γ ≃ γ′, if there is a sequence of unitaries (u_{n}) ∈ ℂI + K ⊗ B such that

• lim_{n→∞} ∥u_{n}γ(a)u_{n}^{∗} − γ′(a)∥ = 0, for all a ∈ A

• u_{n}γ(a)u_{n}^{∗} − γ′(a) ∈ K ⊗ B, for all n, and a ∈ A.

The continuous version of the above equivalence is defined as follows [DE_{2}]. Two representations γ, γ′ : A → 𝕄(K ⊗ B) are called properly asymptotically unitarily equivalent, written γ ≊ γ′, if there is a norm-continuous path of unitaries u : [0, ∞) → ℂI + K ⊗ B such that • lim_{t→∞} ∥u_{t}γ(a)u_{t}^{∗} − γ′(a)∥ = 0, for all a ∈ A

∗The author was partially supported by an NSF grant

• u_{t}γ(a)u_{t}^{∗} − γ′(a) ∈ K ⊗ B, for all t ∈ [0, ∞), and a ∈ A.

As in [DE_{2}], the use of the word ‘proper’ reflects that the unitaries implementing the above equivalence relations are compact perturbations of the identity.

Let A, B be unital C*-algebras and assume that A is separable. For two unital ∗- homomorphisms φ, ψ : A → B consider the following conditions.

(1) [φ] = [ψ] in KK(A, B).

(2) φ ⊕ γ ≊ ψ ⊕ γ for some unital representation γ : A → 𝕄(K ⊗ B).

(3) [φ] = [ψ] in Rørdam’s group KL(A, B) = KK(A, B)/Pext(K_{∗}(A), K_{∗+1}(B)), ([Rø]).

(4) φ ⊕ γ ≃ ψ ⊕ γ for some unital representation γ : A → 𝕄(K ⊗ B).

(5) φ is approximately stably unitarily equivalent to ψ (see Definition 3.7).

By a result of Eilers and the author [DE_{2}] we have (1) ⇔ (2). Note that (2) ⇒ (4) is obvious. Suppose that A is quasidiagonal relative to B. This notion was introduced by Salinas [Sa], see Definition 3.5. Then it is not hard to see that (4) ⇔ (5), see Lemma 3.8.

Condition (3) is stated under the assumption that A satisfies the universal coefficient theorem (abbreviated UCT) of [RS]. In view of the universal multi-coefficient theorem (UMCT) of [DL], (3) is equivalent to

(3’) φ_{∗} = ψ_{∗} : K_{∗}(A, ℤ/n) → K_{∗}(B, ℤ/n) for all n ≥ 0.

In this paper we prove that if A is separable nuclear quasidiagonal relative to B and satisfies the UCT, then (3) ⇔ (4), hence (3)⇔ (5), see Theorem 5.1. The latter equivalence was conjectured informally by Lin [L], and it was known to be true if A is abelian [D_{1}], or if A can be approximated by nuclear C*-subalgebras satisfying the UCT and having finitely generated K-theory and B is simple [L], or if A is simple and B satisfies certain conditions in unstable K-theory and has bounded exponential rank [L], [DE_{1}].

The proof of (3) ⇔ (4) is based on a new proof that we give for the implication (1) ⇒ (5) of [DE_{2}], stated here as Theorem 3.11. Unlike previous approaches, the proof does not use the theorem of Kadison and Ringrose on derivable automorphisms of C*-algebras.

The same result is used to give a KK-theoretical description of the closure of zero in Ext(A, B), see Theorem 4.3. This yields new proofs of several results of Schochet [Sch_{1}]- [Sch_{3}], see Corollaries 4.5–4.7. Corollary 4.7 is used in the proof of Theorem 5.1.

The results on approximate stable unitary equivalence of ∗-homomorphisms have important applications in Elliott’s classification program. There are interesting situations when the maps γ_{n} from Definition 3.7 can be chosen to be ∗-homomorphisms. For instance if A is nuclear and residually finite dimensional then γ_{n} can be taken to be finite dimensional representations of A into matrices over ℂ1_{B}. Another example considered by Lin [L] and generalized in [DE_{1}] is when A is nuclear and there is a full unital embedding ι : A ↪ B. In that case one can take γ_{n} = n · ι = ι ⊕ ⋯ ⊕ ι (n-times).

2 Some preliminaries in KK-theory

Throughout this paper, A is a separable C*-algebra and B is a σ-unital C*-algebra. We work with Hilbert B-modules E countably generated over B such as E = H_{B} = H ⊗ B where H is a separable infinite dimensional Hilbert space. We use the notation from [Kas_{1}]. Let 𝕄(K⊗B) denote the multiplier C*-algebra of K⊗B and let Q(K⊗B) denote the generalized Calkin algebra 𝕄(K ⊗ B)/K ⊗ B. The quotient map 𝕄(K ⊗ B) → Q(K ⊗ B) is denoted by π. Very often we will identify 𝕄(K ⊗ B) with L(H_{B}). Recall that the group Ext^{−1}(A, B) is generated by ∗-homomorphisms from A to Q(K ⊗ B) which admit completely positive liftings A → 𝕄(K ⊗ B). Such a map is called a semisplit extension. The group KK^{1}(A, B) consists of equivalence classes of pairs (τ, P) where τ : A → 𝕄(K ⊗ B) is a representation and P ∈ 𝕄(K ⊗ B) is a selfadjoint projection such that [τ(a), P] ∈ K ⊗ B for all a ∈ A. Kasparov [Kas_{2}] proved that there is an isomorphism κ : KK^{1}(A, B) → Ext^{−1}(A, B) which maps the class of (τ, P) to the class of the extension π ∘ P τ(−)P.

Let x ∈ KK^{1}(C(S^{1}), ℂ) be the element defined by the Toeplitz extension

0 ⟶ K ⟶ T ⟶ C(S^{1}) ⟶ 0.

Recall from [Bla, 17.8.5] that there is a natural homomorphism

τ_{A} : KK^{∗}(C(S^{1}), ℂ) ⟶ KK^{∗}(C(S^{1}) ⊗ A, A).

Using the Kasparov product

KK^{1}(C(S^{1}) ⊗ A, A) × KK(A, B) ⟶ KK^{1}(C(S^{1}) ⊗ A, B)

we define a group homomorphism

τ_{A}(x) ⊗ − : KK(A, B) ⟶ KK^{1}(C(S^{1}) ⊗ A, B).

This homomorphism is injective, and in fact its composition with the restriction map

KK^{1}(C(S^{1}) ⊗ A, B) → KK^{1}(SA, B)

is an isomorphism. It coincides with the homomorphism

τ_{A}(x_{0}) ⊗ − : KK(A, B) ⟶ KK^{1}(SA, B),

where x_{0} ∈ KK^{1}(Sℂ, ℂ) is the element defined by the reduced Toeplitz extension

0 ⟶ K ⟶ T_{0} ⟶ Sℂ ⟶ 0.

Up to a sign, its inverse is given by τ_{A}(y_{0}) ⊗ − : KK^{1}(SA, B) ⟶ KK(A, B), where y_{0} ∈ KK^{1}(ℂ, Sℂ) is the class of the extension

0 ⟶ Sℂ ⟶ Cℂ ⟶ ℂ ⟶ 0.

(Recall that x_{0} ⊗ y_{0} = −1 and y_{0} ⊗ x_{0} = −1 by Bott periodicity, [Bla, 19.2].)

Given two unital ∗-homomorphisms φ, ψ : A → B, we want an explicit computation of τ_{A}(x) ⊗ ([φ] − [ψ]) ∈ KK^{1}(C(S^{1}) ⊗ A, B) or rather of its image in Ext^{−1}(C(S^{1}) ⊗ A, B) under Kasparov’s isomorphism κ : KK^{1}(C(S^{1}) ⊗ A, B) → Ext^{−1}(C(S^{1}) ⊗ A, B). In order to formulate the result we need some notation. Let ℤ_{+} = {k ∈ ℤ : k ≥ 0}, ℤ_{−} = {k ∈ ℤ : k < 0}, and let p_{±} : ℓ^{2}(ℤ) → ℓ^{2}(ℤ_{±}) be the canonical projections. Let S : ℓ^{2}(ℤ) → ℓ^{2}(ℤ) be the bilateral shift, Se_{i} = e_{i+1}, where (e_{i}) is the canonical orthonormal basis of ℓ^{2}(ℤ). Define Φ, Ψ : A → 𝕄(K(ℓ^{2}(ℤ_{+})) ⊗ B) by

Φ = φ ⊕ φ ⊕ ψ ⊕ φ ⊕ ψ ⊕ ⋯ = φ ⊕ (φ ⊕ ψ)_{∞},

Ψ = ψ ⊕ φ ⊕ ψ ⊕ φ ⊕ ψ ⊕ ⋯ = ψ ⊕ (φ ⊕ ψ)_{∞}.

Let w : ℓ^{2}(ℤ_{+}) ⊗ B → ℓ^{2}(ℤ_{+}) ⊗ B be a unitary operator defined by w(e_{0} ⊗ b) = e_{1} ⊗ b, w(e_{2k−1} ⊗ b) = e_{2k+1} ⊗ b and w(e_{2k} ⊗ b) = e_{2k−2} ⊗ b for k ≥ 1 and b ∈ B, where (e_{i}) is the canonical orthonormal basis of ℓ^{2}(ℤ_{+}). Note that Φ(a) = wΨ(a)w^{∗} for all a ∈ A. Therefore [Φ(a), w] ∈ K ⊗ B for all a ∈ A since Φ(a) − Ψ(a) ∈ K ⊗ B.

If x ∈ 𝕄(K ⊗ B), then π(x) ∈ Q(K ⊗ B) will be denoted by ẋ. If σ : A → 𝕄(K ⊗ B) is a map, then π ∘ σ will be denoted by σ̇.

Proposition 2.1 Let A, B be unital C*-algebras with A separable and let φ, ψ : A → B be two unital ∗-homomorphisms. Then with notation as above,

κ(τ_{A}(x) ⊗ [φ] − τ_{A}(x) ⊗ [ψ]) = [σ],

where σ : C(S^{1}) ⊗ A → Q(K ⊗ B) is defined by σ(f ⊗ a) = f(ẇ)Φ̇(a).

Proof. The element τ_{A}(x) ∈ KK^{1}(C(S^{1}) ⊗ A, A) is given by the class of the pair (τ, P_{+}), where τ : C(S^{1}) ⊗ A → 𝕄(K(ℓ^{2}(ℤ)) ⊗ A) = L(ℓ^{2}(ℤ) ⊗ A), τ(f ⊗ a) = f(S) ⊗ a, S being the bilateral shift, and P_{+} = p_{+} ⊗ id_{A}. It is then clear that −τ_{A}(x) is given by the class of (τ, P_{−}) with P_{−} = p_{−} ⊗ id_{A}.

Since φ is a ∗-homomorphism, we have τ_{A}(x) ⊗ [φ] = φ_{∗}(τ_{A}(x)) by [Bla][18.7.2(a)], so that τ_{A}(x) ⊗ [φ] is represented by the class of the pair (τ_{φ}, P_{+}), where τ_{φ} : C(S^{1}) ⊗ A → 𝕄(K(ℓ^{2}(ℤ) ⊗ B), τ_{φ}(f ⊗ a) = f(S) ⊗ φ(a) and P_{+} = p_{+} ⊗ id_{B} : ℓ^{2}(ℤ ⊗ B) → ℓ^{2}(ℤ_{+} ⊗ B).

Similarly, −τ_{A}(x)⊗[ψ] = [τ, P_{−}]⊗[ψ] is represented by (τ_{ψ}, P_{−}), where τ_{ψ} : C(S^{1})⊗A → 𝕄(K(ℓ^{2}(ℤ)) ⊗ B), τ_{ψ}(f ⊗ a) = f(S) ⊗ ψ(a) and P_{−} = p_{−} ⊗ id_{B} : ℓ^{2}(ℤ ⊗ B) → ℓ^{2}(ℤ_{−} ⊗ B). Thus

τ_{A}(x) ⊗ ([φ] − [ψ]) = [τ_{φ}, P_{+}] + [τ_{ψ}, P_{−}] = [τ_{ψ} ⊕ τ_{φ}, P_{−} ⊕ P_{+}].

The image of the latter element in Ext^{−1}(C(S^{1}) ⊗ A, B) under κ is equal to [ω̇], where ω : C(S^{1}) ⊗ A → L(ℓ^{2}(ℤ_{−}) ⊗ B) ⊕ L(ℓ^{2}(ℤ_{+}) ⊗ B) ⊂ L(ℓ^{2}(ℤ) ⊗ B) is a completely positive map defined by ω = P_{−}τ_{ψ}P_{−} + P_{+}τ_{φ}P_{+}. Therefore

ω(f ⊗ a) = p_{−} ⊗ id_{B} (f(S) ⊗ ψ(a)) p_{−} ⊗ id_{B} + p_{+} ⊗ id_{B} (f(S) ⊗ φ(a)) p_{+} ⊗ id_{B}
= p_{−}f(S)p_{−} ⊗ ψ(a) + p_{+}f(S)p_{+} ⊗ φ(a).

Since [f(S), p_{±}] ∈ K, we have

ω(f ⊗ a) − f(S) ⊗ id_{B} (p_{−} ⊗ ψ(a) + p_{+} ⊗ φ(a)) ∈ K(ℓ^{2}(ℤ)) ⊗ B, ∀f ∈ C(S^{1}), a ∈ A.

Therefore, if Φ_{1} = p_{−} ⊗ ψ + p_{+} ⊗ φ = ⋯ ⊕ ψ ⊕ ψ ⊕ φ ⊕ φ ⊕ ⋯, and σ_{1} : C(S^{1}) ⊗ A → Q(K(ℓ^{2}(ℤ) ⊗ B) is defined by σ_{1}(f ⊗ a) = f(π(S ⊗ id_{B}))Φ̇_{1}(a), then σ̇_{1} = ω̇. Let ξ : ℤ → ℤ_{+} be a bijection defined by ξ(0) = 0, ξ(1) = 1, ξ(k) = 2k + 1, if k ≥ 1 and ξ(k) = −k if k < 0. Under the corresponding identification of ℓ^{2}(ℤ) with ℓ^{2}(ℤ_{+}), Φ_{1}, S ⊗ id_{B} correspond to Φ, w, respectively, so that σ_{1} corresponds to σ.

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3 Approximate unitary equivalence

In this section we show that the conditions (1), (4) and (5) from the introduction are related as follows: (1) ⇒ (4) ⇔ (5).

Definition 3.1 Two representations γ : A → L(E), γ′ : A → L(E′) are called approximately unitarily equivalent, written γ ∼ γ′, if there exists a sequence of unitaries (u_{n}) in L(E′, E) such that (i) lim_{n→∞} ∥γ(a) − u_{n}γ′(a)u_{n}^{∗}∥ = 0, a ∈ A,

(ii) γ(a) − u_{n}γ′(a)u_{n}^{∗} ∈ K(E), for all n, a ∈ A.

Definition 3.2 A representation γ : A → L(H_{B}) = 𝕄(K⊗B) is called absorbing if γ ⊕σ ∼ γ for any representation σ : A → L(E). If A is unital, then a representation γ : A → 𝕄(K⊗ B) is called unitally absorbing if γ ⊕ σ ∼ γ for any unital representation σ : A → L(E).

Note that any absorbing representation is injective. Any two absorbing representations are approximately unitarily equivalent.

Examples 3.3 (a) A scalar representation γ : A ⟶ 𝕄(K ⊗ B) is a representation which factors as

A ⟶^{γ′} L(H) ⟶^{−⊗1} L(H) ⊗ 1 ↪ 𝕄(K ⊗ B).

Suppose that A or B are nuclear, γ′ is faithful and γ′(A) ∩ K = {0}. If γ′ is unital, then γ is unitally absorbing. If \overline{γ′(A)H} has infinite codimension in H, then γ is absorbing, [Kas_{2}].

(b) Lin [L] showed that if A is nuclear and separable and if ι : A → B is a unital embedding, then the map d_{ι} : A ⟶ 𝕄(K ⊗ B) defined by d_{ι}(a) = 1 ⊗ ι(a) is unitally absorbing, whenever either A or B is simple. A unital ∗-homomorphism ι : A ↪ B is called a full embedding if the linear span of Bι(a)B is dense in B for all nonzero a ∈ A. It was shown in [DE_{1}] that d_{ι} is unitally absorbing whenever ι is a full embedding.

(c) Thomsen [Tho_{2}] proved the existence of absorbing extensions γ : A → 𝕄(K ⊗ B) for arbitrary separable C*-algebras A and B.

A C*-algebra B is called stably unital if B ⊗ K has a countable approximate unit consisting of projections. A subset E ⊂ 𝕄(K ⊗ B) is called quasidiagonal if there is a countable approximate unit (p_{n}) of K ⊗ B consisting of projections such that

lim_{n→∞} ∥p_{n}a − ap_{n}|| = 0, a ∈ E. (1)

A representation γ : A → 𝕄(K ⊗ B) is called quasidiagonal if the set γ(A) ⊂ 𝕄(K ⊗ B) is quasidiagonal.

Remark 3.4 (a) If γ, γ′ : A → 𝕄(K ⊗ B) are representations with γ ∼ γ′ and γ is quasidiagonal, then γ′ is quasidiagonal.

(b) If A is unital and if γ : A → 𝕄(K ⊗ B) is a unital, unitally absorbing representation, then γ ⊕ 0_{𝕄(K⊗B)} is an absorbing representation [DE_{1}, Lemma 2.17].

(c) Suppose that E ⊂ 𝕄(K ⊗ B) is a quasidiagonal set and that p ∈ E is a projection such that px = xp = x for all x ∈ E. Then there is an approximate unit of projections (p_{n}) of K ⊗ B such that [p_{n}, x] → 0 for all x ∈ E and [p_{n}, p] = 0 for all n. Indeed, since E is a quasidiagonal set, there is an approximate unit of projections (q_{n}) of K ⊗ B such that [q_{n}, x] → 0 for all x ∈ E. In particular [q_{n}, p] → 0. Using functional calculus one can perturb each q_{n} to a projection p_{n} ∈ K ⊗ B, such that ∥p_{n} − q_{n}∥ → 0, and [p_{n}, p] = 0. (see[DE_{1}, Lemma 5.1 (iii)]).

Definition 3.5 [Sa] Let A be a separable C*-algebra and let B be a stably unital C*-algebra. We say that A is quasidiagonal relative to B if there exists an absorbing quasidiagonal representation γ : A → 𝕄(K ⊗ B).

By Remark 3.4 (a) we have that if a separable C*-algebra A is quasidiagonal relative to B, then any absorbing representation γ : A → 𝕄(K ⊗ B) is quasidiagonal. It also follows from Remark 3.4 that a separable unital C*-algebra is quasidiagonal relative to a stably unital C*-algebra B if and only if any (or some) unital unitally absorbing representation is quasidiagonal. Note that a C*-algebra A is quasidiagonal if and only if is quasidiagonal relative to ℂ.

Lemma 3.6 Let C be a unital nuclear C*-subalgebra of a UHF algebra. Let A be a unital separable C*-algebra and let B be a stably unital C*-algebra. If A is quasidiagonal relative to B, then C ⊗ A is quasidiagonal relative to B.

Proof. By assumption, there is a unital embedding ȷ : C ↪ D where D is a unital UHF algebra D = \overline{∪D_{i}} with D_{i} ≅ M_{r(i)}(ℂ). Let π : D → L(H) and σ : A → 𝕄(K ⊗ B) be unital unitally absorbing representations. It is clear that π⊗σ : D⊗A → L(H)⊗L(H_{B}) ⊂ 𝕄(K⊗B) is quasidiagonal since both π and σ are so. Therefore πȷ⊗σ is a quasidiagonal representation of C ⊗ A. It remains to prove that πȷ ⊗ σ is unitally absorbing. First we observe that π ⊗ σ is unitally absorbing since its restriction to each of the D_{i} ⊗ A ≅ M_{r(i)}(ℂ) ⊗ A is easily seen to be unitally absorbing, using the assumption that σ is unitally absorbing. 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

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

• lim_{i→∞} ∥v_{i}^{∗}ξ∥ = 0 for all ξ ∈ E.

Here E, F are countable B-modules. As a corollary of [DE_{1}, Theorem 2.13] a unital representation γ : A → 𝕄(K ⊗ B) is unitally absorbing if and only if φ ≺ γ for any completely positive contraction φ : A → K(F). Therefore in order to prove that πȷ ⊗ σ is unitally absorbing it will suffice to show that φ ≺ πȷ ⊗ σ for any completely positive contraction φ : C ⊗ A → K(F).

Since C is nuclear, we find two sequences of completely positive contractions

α_{n} : C → M_{k(n)}(ℂ), β_{n} : M_{k(n)}(ℂ) → C

such that ∥β_{n}α_{n}(c) − c∥ → 0 for all c ∈ C. By Arveson’s extension theorem, we can extend α_{n} to a completely positive contraction α′_{n} : D → M_{k(n)}(ℂ) such that α′_{n} ∘ ȷ = α_{n}. If E_{n} = β_{n} ∘ α′_{n} : D → C, then E_{n} is a completely positive contraction with ∥E_{n}ȷ(c) − c∥ → 0 for all c ∈ C. Since π ⊗ σ is unitally absorbing, we have φ ∘ (E_{n} ⊗ id_{A}) ≺ π ⊗ σ, hence

φ_{n} = φ ∘ (E_{n} ⊗ id_{A}) ∘ (ȷ ⊗ id_{A}) = φ ∘ (E_{n}ȷ ⊗ id_{A}) ≺ πȷ ⊗ σ.

Since ∥φ_{n}(x) − φ(x)∥ → 0 for all x ∈ C ⊗ A, it follows that φ ≺ πȷ ⊗ σ.

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Definition 3.7 Let A, B be unital C*-algebras and assume that A is quasidiagonal relative to B. Two unital ∗-homomorphisms φ, ψ : A → B are called approximately stably unitarily equivalent if for any unital unitally absorbing representation γ : A → 𝕄(K ⊗ B), and any approximate unit of projections (p_{n}) of K ⊗ B with ∥[γ(a), p_{n}]∥ → 0, a ∈ A, if γ_{n}(a) = p_{n}γ(a)p_{n}, then there exist an increasing sequence of integers (k(n)) and a sequence of partial isometries (v_{n}) in K ⊗ B such that v_{n}^{∗}v_{n} = v_{n}v_{n}^{∗} = 1_{B} ⊕ p_{k(n)} and

lim_{n→∞} ∥v_{n}(φ(a) ⊕ γ_{k(n)}(a))v_{n}^{∗} − ψ(a) ⊕ γ_{k(n)}(a)|| = 0, a ∈ A. (2)

Lemma 3.8 Let φ, ψ : A → B be two unital ∗-homomorphisms. Suppose that A is separable and quasidiagonal relative to B. Then the following conditions are equivalent.

(i) φ ⊕ η ≃ ψ ⊕ η for some unital representation η : A → 𝕄(K ⊗ B).

(ii) φ ⊕ γ ≃ ψ ⊕ γ for some unital unitally absorbing representation γ : A → 𝕄(K ⊗ B).

(iii) φ ⊕ γ ≃ ψ ⊕ γ for any unital unitally absorbing representation γ : A → 𝕄(K ⊗ B).

(iv) φ is approximately stably unitarily equivalent to ψ.

(v) φ and ψ satisfy the conditions of Definition 3.7 for some γ and some (p_{n}).

Proof. The implications (iii) ⇒ (ii) ⇒ (i) and (iv) ⇒ (v) are obvious. By [DE_{2}, Lemma 3.4] if γ ∼ γ′ then φ ⊕ γ ≃ ψ ⊕ γ if and only if φ ⊕ γ′ ≃ ψ ⊕ γ′. This proves that (ii) ⇒ (iii). To prove (i) ⇒ (ii), note that (i) ⇒ φ ⊕ η ⊕ γ ≃ ψ ⊕ η ⊕ γ. This readily implies (ii) since η ⊕ γ ∼ γ as γ is absorbing.

(iii) ⇒ (iv) Let γ : A → 𝕄(K ⊗ B) and (p_{n}) be as in Definition 3.7. By assumption there is a sequence of unitaries (u_{n}) in ℂI + K ⊗ B such that

u_{n}(φ(a) ⊕ γ(a))u_{n}^{∗} − ψ(a) ⊕ γ(a) → 0, a ∈ A (3)

as n → ∞. We also have ∥p_{n}γ(a) − γ(a)p_{n}∥ → 0, a ∈ A. After passing to a subsequence (p_{k(n)}) we may arrange that ∥[e_{n}, u_{n}]∥ → 0 as n → ∞, where e_{n} = 1_{B} ⊕ p_{k(n)}. By functional calculus we find a sequence of unitaries v_{n} ∈ e_{n}(K ⊗ B)e_{n} such that ∥e_{n}u_{n}e_{n} − v_{n}∥ → 0 as n → ∞. Compressing by e_{n} in (3) we obtain that

∥v_{n}(φ(a) ⊕ γ_{k(n)}(a))v_{n}^{∗} − ψ(a) ⊕ γ_{k(n)}(a)∥ → 0

as n → ∞ for all a ∈ A. This proves (iv).

(v) ⇒ (ii) Fix γ : A → 𝕄(K ⊗ B), (p_{n}), (k(n)) and (v_{n}) as in Definition 3.7. We want to prove that φ ⊕ γ ≃ ψ ⊕ γ. Let u_{n} = 0_{B} ⊕ (1 − p_{k(n)}) + v_{n} ∈ ℂI + K ⊗ B and define γ′_{k(n)}(a) = (1 − p_{k(n)})γ(a)(1 − p_{k(n)}). We have

∥γ(a) − γ_{k(n)}(a) − γ′_{k(n)}(a)∥ → 0, as n → ∞, a ∈ A, (4)

since [γ(a), p_{n}] → 0. It is now clear that (3) follows from (4) and (2). This proves that φ ⊕ γ ≃ ψ ⊕ γ.

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We regard Q(K ⊗ B) ⊕ Q(K ⊗ B) as a unital subalgebra of Q(K ⊗ B) is the usual way.

Lemma 3.9 Let A be a unital separable C*-algebra and let B be a stably unital C*-algebra. Suppose that A is quasidiagonal relative to B and let σ be a unital semisplit extension such that [σ] = 0 in Ext^{−1}(A, B). Then for any unital unitally absorbing representation γ : A → 𝕄(K ⊗ B) the set

E_{σ⊕γ̇} = {X ∈ 𝕄(K ⊗ B) : Ẋ ∈ (σ ⊕ γ̇)(A)}

is quasidiagonal.

Proof. Since γ is unitally absorbing, γ ⊕ 0_{𝕄(K⊗B)} is absorbing by Remark 3.4 (b). Since [σ] = 0 in Ext^{−1}(A, B), σ⊕γ̇ ⊕0 is of the form u̇(γ̇ ⊕0⊕0)u̇^{∗} for some unitary u ∈ 𝕄(K⊗B). Therefore it lifts to an absorbing representation δ : A → 𝕄(K⊗B). Since A is quasidiagonal relative to B, δ(A)+K⊗B is a quasidiagonal set. Finally we observe that E_{σ⊕γ̇} ⊕0_{𝕄(K⊗B)} ⊂ δ(A) + K ⊗ B, so that E_{σ⊕γ̇} is quasidiagonal by Remark 3.4(c), as it contains an element acting as a unit.

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Lemma 3.10 Let A, B be unital C*-algebras with A separable. Let φ, ψ : A → B be two unital ∗-homomorphisms. Suppose that there exist a unital unitally absorbing representation η : A → 𝕄(K ⊗ B) and a unitary u ∈ 𝕄(K ⊗ B) such that

(i) φ ⊕ η = u(ψ ⊕ η)u^{∗}

(ii) the set (φ ⊕ η)(A) ∪ {u} is quasidiagonal in 𝕄(K ⊗ B).

Then φ is approximately stably equivalent to ψ.

Proof. From (ii) there exists an approximate unit of projections (e_{n}) of K ⊗ B such that for all a ∈ A

[e_{n}, (φ ⊕ η)(a)] → 0, [e_{n}, u] → 0 as n → ∞. (5)

From (i) and (5) [e_{n}, (ψ ⊕ η)(a)] → 0 as n → ∞. We may also arrange that, in addition to the above, e_{n} ≥ e_{11} ⊗ 1_{B} ∈ K ⊗ B, so that e_{n} = e_{11} ⊗ 1_{B} + q_{n} where (q_{n}) is a sequence of projections satisfying [q_{n}, η(a)] → 0 for all a ∈ A as n → ∞. Compressing by e_{n} in (i), we obtain that for all a ∈ A

∥φ(a) ⊕ q_{n}η(a)q_{n} − e_{n}ue_{n}(ψ(a) ⊕ q_{n}η(a)q_{n})e_{n}u^{∗}e_{n}∥ → 0, as n → ∞. (6)

Since [e_{n}, u] → 0, for large n, u_{n} = e_{n}ue_{n}|e_{n}ue_{n}|^{−1/2} is a unitary in e_{n}(K ⊗ B)e_{n} with ∥u_{n} − e_{n}ue_{n}∥ → 0. We then obtain from (6) that

∥φ(a) ⊕ q_{n}η(a)q_{n} − u_{n}(ψ(a) ⊕ q_{n}η(a)q_{n})u_{n}^{∗}∥ → 0, as n → ∞.

We conclude the proof by applying (v) ⇒ (iv) of Lemma 3.8.

Theorem 3.11 ([DE_{2}]) Let A, B be unital C*-algebras. Assume that A is separable, and quasidiagonal relative to B. Let φ, ψ : A → B be two unital ∗-homomorphisms with [φ] = [ψ] in KK(A, B). Then φ is approximately stably equivalent to ψ. Equivalently, φ ⊕ γ ≃ ψ ⊕ γ, for any unital unitally absorbing representation γ : A → 𝕄(K ⊗ B).

Proof. We are going to find a representation η : A → 𝕄(K⊗B) and a unitary u ∈ 𝕄(K⊗B) such that φ, ψ, η and u satisfy the assumptions of Lemma 3.10. Recall that the canonical map 𝕄(K⊗B) → Q(K⊗B) is denoted by π and that we sometimes write ȧ for π(a). We are using the notation from Proposition 2.1. Since τ_{A}(x) ⊗ − : KK(A, B) → KK^{1}(C(S^{1}) ⊗ A, B) is a homomorphism and [φ] − [ψ] = 0, it follows from Proposition 2.1 that [σ] = 0 in Ext^{−1}(C(S^{1}) ⊗ A, B). By assumption, A is quasidiagonal relative to B. Since C(S^{1}) embeds unitally in a UHF algebra, it follows from Lemma 3.6 that C(S^{1}) ⊗ A is also quasidiagonal relative to B. Therefore there exists a unital, unitally absorbing quasidiagonal representation Δ : C(S^{1}) ⊗ A → 𝕄(K ⊗ B). By adding a suitable representation to Δ, if necessary, we may arrange that if δ is the restriction of Δ to 1 ⊗ A, δ(a) = Δ(1 ⊗ a), then δ is also unitally absorbing. Let v = Δ(z ⊗ 1), where z is the canonical unitary generator of C(S^{1}). Recall that

Φ = φ ⊕ (φ ⊕ ψ)_{∞} Ψ = ψ ⊕ (φ ⊕ ψ)_{∞}, Φ = wΨw^{∗}.

Therefore if we set η = (φ ⊕ ψ)_{∞} ⊕ δ, and u = w ⊕ v, then

φ ⊕ η = Φ ⊕ δ, ψ ⊕ η = Ψ ⊕ δ, and φ ⊕ η = u(ψ ⊕ η)u^{∗}

since Φ = wΨw^{∗} and δ commutes with v. Thus condition (i) of Lemma 3.10 satisfied. Note that η is unital unitally absorbing since its direct summand δ is so. It remains to verify condition (ii) of the same lemma. With our notation, that amounts to checking that the set

E = (Φ ⊕ δ)(A) ∪ {u}

is quasidiagonal. Note that E ⊂ E_{σ⊕Δ̇} since π(Φ ⊕ δ) = σ|_{1⊗A} ⊕ δ̇ and u̇ = ẇ ⊕ v̇ = (σ ⊕ Δ̇)(z ⊗ 1). Now E_{σ⊕Δ̇} is quasidiagonal by Lemma 3.9, since [σ] = 0 in Ext^{−1}(A, B). Therefore its subset E is quasidiagonal as well.

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Corollary 3.12 ([DE_{1}]) Let A, B be unital C*-algebras and let ι : A → B be a full unital embedding (Example 3.3(b)). Suppose that A is nuclear and separable. Let φ, ψ : A ⟶ B be two unital ∗-homomorphisms such that [φ] = [ψ] in KK(A, B). Then for any finite subset F ⊆ A and any ϵ > 0, there exist n and a unitary u ∈ M_{n+1}(B) such that

∥(u(φ(a) ⊕ n · ι(a))u^{∗} − ψ(a) ⊕ n · ι(a)∥ < ϵ.

for all a ∈ F.

Proof. This follows from Theorem 3.11 applied for γ = d_{ι}.

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4 The closure of zero in Ext(A, B)

In this section we give a purely KK-theoretical description of the closure of zero in Ext(A, B) under the assumption that A is separable and nuclear and B is σ-unital.

The following proposition is well known to the specialists. A proof is included for the sake of completeness.

Proposition 4.1 If A is a separable C*-algebra then KK^{i}(A, 𝕄(K ⊗ B)) = 0, i = 0, 1, for any C*-algebra B.

Proof. This is similar to the proof of [D_{2}, Proposition 2.2]. To simplify the notation, let M = 𝕄(K ⊗ B) = L(H_{B}). Since KK^{1}(A, M) ≅ KK(SA, M) is suffices to consider only the case i = 0. Since M is unital, KK(A, M) ≅ [qA, K(E) ⊗ M], by [Cun], where E is an infinite dimensional Hilbert space. The addition on the latter group is given by [φ] + [ψ] = [θ_{φ,ψ}], θ_{φ,ψ}(a) = w_{1}φ(a)w_{1}^{∗} + w_{2}ψ(a)w_{2}^{∗}, where w_{i} ∈ M(K(E) ⊗ M) are isometries with w_{1}w_{1}^{∗}+w_{2}w_{2}^{∗} = 1. This definition does not depend on the particular choice of the pair w_{i} (see [JT, 1.3]). Let s_{1}, s_{2} : H_{B} → H_{B} be isometries with s_{1}s_{1}^{∗} +s_{2}s_{2}^{∗} = 1 and set w_{i} = 1_{E} ⊗s_{i} : E ⊗ H_{B} → E ⊗ H_{B}. Let s : H_{B} ⊕ H_{B} → H_{B} be defined by s(x_{1} ⊕ x_{2}) = s_{1}x_{1} + s_{2}x_{2}.Then 1_{E}⊗s : E⊗H_{B}⊕E⊗H_{B} → E⊗H_{B} is a unitary such that θ_{φ,ψ}(a) = 1_{E}⊗s(φ(a)⊕ψ(a))1_{E}⊗s^{∗}.

Define α : 𝕄(K ⊗ B) → 𝕄(K ⊗ K ⊗ B) by α(x) = 1_{H} ⊗ x and let v = v_{0} ⊗ id_{B} : H ⊗ H ⊗ B → H ⊗ B for some unitary v_{0} : H ⊗ H → H.

If φ : qA → K(E) ⊗ M is a ∗-homomorphism, define η : qA → K(E) ⊗ M by

η(a) = 1_{E} ⊗ v(id_{K(E)} ⊗ α) ∘ φ(a))1_{E} ⊗ v^{∗}, a ∈ A. (7)

We are going to show that [φ] + [η] = [η], or equivalently [θ_{φ,η}] = [η], in [qA, K(E) ⊗ M]. That will show that [φ] = 0 for all φ so that [qA, K(E) ⊗ M] = 0. Since the unitary group of 𝕄(K(E) ⊗ M) is path connected in the strict topology [JT], it suffices to find a unitary w in 𝕄(K(E)⊗M) such that wθ_{φ,η}w^{∗} = η. Let (e_{i}), i = 0, 1, ... be an orthonormal basis of H and define a unitary u_{0} : H ⊕H ⊗H → H ⊗H by u_{0}(h⊕0) = e_{0} ⊗h, u_{0}(e_{i} ⊗h) = e_{i+1} ⊗h, h ∈ H. Set u = u_{0} ⊗id_{B} : H ⊗B ⊕H ⊗H ⊗B → H ⊗H ⊗B and note that 1_{H} ⊗x = u(x⊕1_{H} ⊗x)u^{∗} for all x ∈ M, hence α = u(id_{M} ⊕ α)u^{∗}. Therefore

(id_{K(E)} ⊗ α) ∘ φ(a) = 1_{E} ⊗ u(φ(a) ⊕ (id_{K(E)} ⊗ α) ∘ φ(a))1_{E} ⊗ u^{∗}, a ∈ A. (8)

Finally, using (7) and (8), one checks that

w = 1_{E} ⊗ (vu(1_{H_{B}} ⊕ v^{∗})s^{∗}) ∈ 1_{E} ⊗ M ⊂ 𝕄(K(E) ⊗ M)

is a unitary satisfying wθ_{φ,η}w^{∗} = η. Note that the unitary vu(1_{H_{B}}⊕v^{∗})s^{∗} belongs to 𝕄(K⊗B) since it is given by a composite of B-linear unitaries

H_{B} ⟶^{s^{∗}} H_{B} ⊕ H_{B} ⟶^{1⊕v^{∗}} H_{B} ⊕ (H ⊗ H)_{B} ⟶^{u} (H ⊗ B)_{B} ⟶^{v} H_{B},

where (H ⊗ H)_{B} = H ⊗ H ⊗ B.

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For the remaining of this section we assume that A is a separable nuclear C*-algebra and B is σ-unital. In this case all extensions σ : A → Q(K ⊗ B) are semisplit by the Choi-Effros theorem, and Ext^{−1}(A, B) = Ext(A, B). Any ∗-homomorphism φ : A → Q(K ⊗ B) gives both and element [φ]_{Ext} of Ext(A, B) and element [φ]_{KK} of KK(A, Q(K ⊗ B)).

Proposition 4.2 Let A be a separable nuclear C*-algebra and let B be a σ-unital C*-algebra. Then the map χ : Ext(A, B) → KK(A, Q(K ⊗ B)), defined by χ[φ]_{Ext} = [φ]_{KK}, is a natural isomorphism of groups.

Proof. From the six-term exact sequence for KK(A, −) associated with the extension

0 → K ⊗ B → 𝕄(K ⊗ B) → Q(K ⊗ B) → 0 (9)

and from Proposition 4.1 we get an exact sequence

0 = KK(A, 𝕄(K ⊗ B)) → KK(A, Q(K ⊗ B)) →^{∂} KK^{1}(A, K ⊗ B) → KK^{1}(A, 𝕄(K ⊗ B)) = 0. (10)

Thus ∂ is an isomorphism. By a theorem of Kasparov [Kas_{2}], there is a natural isomorphism

κ : KK^{1}(A, B) ⟶ Ext(A, B) = Ext(A, B)^{−1}.

Therefore ∂^{−1} ∘ κ^{−1} : Ext(A, B) → KK(A, Q(K ⊗ B)) is an isomorphism. We need to show that ∂^{−1} ∘ κ^{−1} = χ. The image of [φ]_{Ext} under κ^{−1} is denoted by δ_{φ}. Let

0 ⟶ K ⊗ B ⟶ E_{φ} ⟶ A ⟶ 0 (11)

be the pullback of the extension (9) by φ : A → Q(K ⊗ B). Then we have a commutative diagram

0 ⟶ K ⊗ B ⟶ 𝕄(K ⊗ B) ⟶^{π} Q(K ⊗ B) ⟶ 0
∥  ↑  ↑^{φ}
0 ⟶ K ⊗ B ⟶  E_{φ}  ⟶  A  ⟶ 0

By the naturality of the boundary map ∂ we obtain a commutative diagram:

KK(A, Q(K ⊗ B)) ⟶^{∂} KK^{1}(A, K ⊗ B)
φ_{∗}↑  ∥
KK(A, A)  ⟶^{∂} KK^{1}(A, K ⊗ B)

Therefore

∂φ_{∗}[id_{A}]_{KK} = ∂[id_{A}]_{KK}.

Now φ_{∗}[id_{A}]_{KK} = ∂[φ]_{KK} and by [Bla, Theorem 19.5.7]

∂[id_{A}]_{KK} = [id_{A}] ⊗ δ_{φ} = δ_{φ} = κ^{−1}[φ]_{Ext}.

Therefore ∂[φ]_{KK} = κ^{−1}[φ]_{Ext}, hence χ = ∂^{−1} ∘ κ^{−1}. This completes the proof as we have seen that both ∂ and κ are natural isomorphisms.

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Following [BDF] and [Br], Salinas [Sa] introduced a natural topology on Ext(A, B). This is just the quotient topology of the point-norm topology on the space of extensions A → Q(K ⊗ B). An extension τ is called absorbing if for any trivial extension σ, τ ⊕ σ is unitarily equivalent to τ via a unitary liftable to 𝕄(K ⊗ B). If γ : A → 𝕄(K ⊗ B) is a scalar absorbing representation (see Example 3.3(a)), then θ = γ̇ is an absorbing extension. Also τ ⊕ θ is absorbing for any extension τ. If τ : A → Q(K ⊗ B) is an absorbing extension, and if θ : A → Q(K ⊗ B) is a trivial absorbing extension, then [τ] belongs to the closure of zero in Ext^{−1}(A, B), denoted by Z(A, B), if and only if there is a sequence of unitaries u_{n} ∈ Q(K ⊗ B) such that ∥τ(a) − u_{n}θ(a)u_{n}^{∗}∥ → 0 for all a ∈ A, as n → ∞. Since both τ and θ are absorbing, one can arrange that the unitaries u_{n} lift to unitaries in 𝕄(K ⊗ B). This is easily seen if we keep in mind that τ, θ are unitarily equivalent with τ ⊕ 0, θ ⊕ 0, respectively, via liftable unitaries.

To simplify notation, Q(K ⊗ B) will be denoted by Q. Let d : Q → ∏_{n=1}^{∞} Q be defined by d(x) = (x, x, x, …). Let

ν : ∏_{n=1}^{∞} Q → ∏_{n=1}^{∞} Q/ ∑_{n=1}^{∞} Q

denote the quotient map and let

Ω : KK(A, Q) → KK(A, ∏ Q/ ∑ Q)

be the map induced by ν∘d, that is Ω = (ν∘d)_{∗}. Fix θ = γ̇ : A → Q where γ : A → 𝕄(K⊗B) is an absorbing scalar representation as in Example 3.3(a).

Theorem 4.3 Let A, B be C*-algebras with A nuclear and separable. Then Ker Ω consists of the classes [φ]_{KK} of those ∗-homomorphisms φ : A → Q(K ⊗ B) for which there exists a sequence of unitaries (u_{n}) in M_{2}(Q(K ⊗ B)) such that for all a ∈ A

lim_{n→∞} ∥u_{n}(φ ⊕ θ)(a)u_{n}^{∗} − (θ ⊕ θ)(a)∥ = 0. (12)

Equivalently, if φ : A → Q(K ⊗ B) is an absorbing extension, then [φ] ∈ Ker Ω if and only if there is a sequence of unitaries v_{n} ∈ 𝕄(K ⊗ B) such that lim_{n→∞} ∥φ(a) − v̇_{n}θ(a)v̇_{n}^{∗}∥ = 0 for all a ∈ A.

Proof. If φ : A → Q is a ∗-homomorphism and θ is as above, we let

Φ = ν ∘ d ∘ φ, Θ = ν ∘ d ∘ θ.

Suppose that φ and θ satisfy the condition (12) and let us show that Ω[φ]_{KK} = 0. The sequence (u_{n}) gives a unitary u ∈ M_{2}(∏ Q/ ∑ Q) such that u(Φ ⊕ Θ)u^{∗} = Θ ⊕ Θ. This clearly implies that [Φ] = [Θ] in KK(A, ∏ Q/ ∑ Q). On the other hand

[Θ]_{KK} = Ω[θ]_{KK} = Ωχ[θ]_{Ext} = 0,

since [θ]_{Ext} = 0 as θ lifts to a representation γ.

Conversely, assume that Ω[φ]_{KK} = 0. Let us observe that since θ is absorbing, for any integer m ≥ 1, m · θ = θ ⊕ ⋯ ⊕ θ (m − times) is unitarily equivalent to θ. Therefore the condition (12) is equivalent to the following:

For any finite subset F of A and any ϵ > 0, there exist m ≥ 1 and a unitary u ∈ M_{m+1}(Q) such that

∥u(φ ⊕ m · θ)(a)u^{∗} − (θ ⊕ m · θ)(a)∥ < ϵ, a ∈ F. (13)

Consequently, it suffices to prove that φ and θ satisfy (13) rather than (12). Since Ω[φ] = 0 by assumption, and Ω[θ] = 0 as we saw above, we have Ω[φ]_{KK} = Ω[θ]_{KK}, hence [Φ] = [Θ] in KK(A, ∏ Q/ ∑ Q).

Let Ã denote the C*-algebra obtained by adding a unit to A. This is done even if A was unital in the first place. By replacing φ by φ⊕θ if necessary, we may arrange that 1_{Q} ∉ φ(A). Let φ̃, θ̃: Ã → Q be the unital extensions of φ and θ and set Φ̃ = ν ∘ d ∘ φ̃ and Θ̃ = ν ∘ d ∘ θ̃. We have that [Φ̃] = [Θ̃] in KK(Ã, ∏ Q/ ∑ Q) since [Φ] = [Θ] in KK(A, ∏ Q/ ∑ Q), and Φ̃, Θ̃ are unitizations of Φ, Θ.

Note that Θ̃ : A → ∏ Q/ ∑ Q is a full embedding since it factors as a product of unital maps

A \underrightarrow{γ̃′} L(H) ↪ 𝕄(K ⊗ B) → Q(K ⊗ B) ⟶ ∏ Q/ ∑ Q

and γ̃ is a full embedding since γ̃′(A) ∩ K(H) = {0}. By Corollary 3.12, for any finite subset F of A and any ϵ > 0, there exist m ≥ 1 and a unitary U ∈ M_{m+1}(∏ Q/ ∑ Q) such that

∥U(Φ̃ ⊕ m · Θ̃)(a)U^{∗} − (Θ̃ ⊕ m · Θ̃)(a)∥ < ϵ, a ∈ F. (14)

Let V = (v_{i}) ∈ ∏ Q be a unitary lifting of U. Then it follows from (14) that there is some large i such that

∥v_{i}(φ̃ ⊕ m · θ̃)(a)v_{i}^{∗} − (θ̃⊕ m · θ̃)(a)∥ < ϵ, a ∈ F. (15)

This shows that φ and θ satisfy (13) and completes the proof.

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Remark 4.4 The map Ω can be integrated in a six-term exact sequence

KK(A, Q)  ⟶^{Ω} KK(A, ∏ Q/ ∑ Q) ⟶^{1−σ} KK(A, ∏ Q/ ∑ Q)
↑  ↓
KK^{1}(A, ∏ Q/ ∑ Q) ⟵^{1−σ} KK^{1}(A, ∏ Q/ ∑ Q) ⟵^{Ω}  KK^{1}(A, Q)

which is similar to an exact sequence in E-theory found by Thomsen [Tho_{1}].

Note that Theorem 4.3 says that χ(Z(A, B)) = Ker Ω. Let α ∈ KK(A, A′) be a KKequivalence. If Q = Q(K ⊗ B) as before, α induces a commutative diagram:

Ext(A, B) ⟶^{κ^{−1}} KK^{1}(A, B) ⟶^{∂^{−1}} KK(A, Q) ⟶^{Ω} KK(A, ∏ Q/ ∑ Q)
α⊗−↓  ^{α⊗−}↓  ^{α⊗−}↓  ^{α⊗−}↓
Ext(A′, B) ⟶^{κ^{−1}} KK^{1}(A′, B) ⟶^{∂^{−1}} KK(A′, Q) ⟶^{Ω′} KK(A′, ∏ Q/ ∑ Q)

This gives right away the following corollary.

Corollary 4.5 ([Sch_{1}]) Let A, A′ be separable nuclear C*-algebras and let B be a σ-unital C*-algebra. If α ∈ KK(A, A′) is a KK-equivalence, then the isomorphism α⊗− : Ext(A, B) → Ext(A′, B) maps Z(A, B) onto Z(A′, B).

Corollary 4.6 ([Sch_{2}]) Let A be a separable nuclear C*-algebra satisfying the UCT and let B be a σ-unital C*-algebra. Then Z(A, B) is naturally isomorphic to Pext(K_{∗}(A), K_{∗}(B)).

Proof. If A satisfies the UCT, then A is KK-equivalent to a commutative C*-algebra. Therefore A satisfies the UMCT of [DL], so that there is an exact sequence

0 → Pext(K_{∗}(A), K_{∗+1}(Q(K ⊗ B)) → KK(A, Q(K ⊗ B)) → Hom_{Λ}(\underline{K}(A), \underline{K}(Q(K ⊗ B))) → 0, (16)

where \underline{K}(A) = ⊕_{n}K_{∗}(A; ℤ/n) is the total K-theory group, and Λ is a certain set of operations on \underline{K}(−). We regard both Z(A, B) ≅ χ(Z(A, B)) and Pext(K_{∗}(A), K_{∗}(B)) ≅ Pext(K_{∗}(A), K_{∗+1}(Q(K ⊗ B))) as subgroups of KK(A, Q(K ⊗ B)). First we want to verify the inclusion

Z(A, B) ⊂ Pext(K_{∗}(A), K_{∗+1}(Q(K ⊗ B)) ).

If θ is a trivial absorbing extension, then [θ] = 0 so that the map \underline{θ} : \underline{K}(A) → \underline{K}(Q(K ⊗ B)) vanishes. If [σ] ∈ Z(A, B) is the class of an absorbing extension σ, then by the very definition of the topology on Ext(A, B), σ is approximately unitarily equivalent to θ, hence \underline{σ} = \underline{θ} = 0. Therefore [σ] = [σ] − [θ] ∈ Pext(K_{∗}(A), K_{∗+1}(Q(K ⊗ B)) ) by the UMCT (16). To prove the opposite inclusion

Pext(K_{∗}(A), K_{∗+1}(Q(K ⊗ B)) ) ⊂ Z(A, B),

we note that since both subsets are invariant under KK-equivalence in the first variable, it suffices to prove the statement for any C*-algebra KK-equivalent to A. Thus, using [RS, Proposition 7.3], we may assume that A = ∪_{i=1}^{∞}A_{i} where A_{i} are nuclear C*-algebras satisfying the UCT and such that K_{∗}(A_{i}) is finitely generated for each i. In particular Pext(K_{∗}(A_{i}), K_{∗+1}(Q(K ⊗ B)) ) = 0 for all i. Let σ, θ : A → Q(K ⊗ B) be two absorbing extensions with θ trivial and [σ] ∈ Pext(K_{∗}(A), K_{∗+1}(Q(K ⊗ B)). Then [σ|_{A_{i}}] ∈ Pext(K_{∗}(A_{i}), K_{∗+1}(Q(K ⊗ B))) vanishes, so that [σ|_{A_{i}}] = [θ|_{A_{i}}] ∈ KK(A_{i}, Q(K ⊗ B)) by (16). Since both σ|_{A_{i}} and θ|_{A_{i}} are absorbing, they are unitarily equivalent. Therefore σ is approximately unitarily equivalent to θ, hence [σ] ∈ Z(A, B).

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Corollary 4.7 ([Sch_{3}]) Let A be a separable nuclear C*-algebra satisfying the UCT, and let B be a stably-unital C*-algebra. Suppose that A is quasidiagonal relative to B. Let σ : A → Q(K ⊗ B) be an absorbing extension. Then E_{σ} = {y ∈ 𝕄(K ⊗ B) : ẏ ∈ σ(A)} is a quasidiagonal set if and only if [σ] ∈ Pext(K_{∗}(A), K_{∗}(B)).

Proof. This follows from Corollary 4.6 and from [Sa, Theorem 4.4] which states that if A is quasidiagonal relative to B and σ : A → Q(K ⊗ B) is an absorbing extension, then E_{σ} is a quasidiagonal set if and only if [σ] ∈ Z(A, B). It should be noted that the proof from [Sa] also applies to the case when A is nonunital. ■

5 Approximate unitary equivalence revisited

In this section we prove the equivalences (3)⇔ (4) ⇔ (5) from the introduction.

Theorem 5.1 Let A, B be unital C*-algebras with A nuclear and separable. Suppose that A is quasidiagonal relative to B and let φ, ψ : A → B be two unital ∗-homomorphisms. The following assertions are equivalent.

(i) [φ] − [ψ] ∈ Pext(K_{∗}(A), K_{∗+1}(B)) in KK(A, B).

(ii) φ ⊕ γ ≃ ψ ⊕ γ for some (any) unital unitally absorbing quasidiagonal representation γ : A → 𝕄(K ⊗ B).

(iii) φ is approximately stably unitarily equivalent to ψ.

Proof. We have that (ii) ⇔ (iii) by Lemma 3.8.

(iii) ⇒ (i) It follows easily from (iii) that φ_{∗} = ψ_{∗} : K_{∗}(A, ℤ/n) → K_{∗}(B, ℤ/n) for all n ≥ 0, so that (i) follows from the UMCT (16).

(i) ⇒ (iii) Let T be the group morphism defined as the composition

KK(A, B) ⟶^{τ_{A}(x)⊗−} KK^{1}(C(S^{1}) ⊗ A, B) ⟶^{∂^{−1}} KK(C(S^{1}) ⊗ A, Q(K ⊗ B)),

with τ_{A}(x) as in Proposition 2.1 and ∂^{−1} as in Theorem 4.3. The morphism T is clearly compatible with the UMCT (16), in the sense that it induces a commutative diagram

KK(A, B)  ⟶  Hom_{Λ}(\underline{K}(A), \underline{K}(Q(K ⊗ B)))
T ↓  \underline{T} ↓
KK(C(S^{1}) ⊗ A, B) ⟶ Hom_{Λ}(\underline{K}(C(S^{1}) ⊗ A), \underline{K}(Q(K ⊗ B)))

Here \underline{T} (h) = ∂_{∗}^{−1} ∘ h ∘ τ, where τ : \underline{K}(C(S^{1}) ⊗ A) → \underline{K}_{+1}(A) is induced by τ_{A}(x) ∈ KK^{1}(C(S^{1}) ⊗ A, A) and ∂_{∗}^{−1} is the inverse of the isomorphism ∂_{∗} : \underline{K}(Q(K ⊗ B)) → \underline{K}_{+1}(B). That shows that T maps Pext(K_{∗}(A), K_{∗+1}(B)) to Pext(K_{∗}(C(S^{1}) ⊗ A), K_{∗+1}(Q(K ⊗ B)). Note that

T = ∂^{−1} ∘ (τ_{A}(x) ⊗ −) = χ ∘ κ ∘ (τ_{A}(x) ⊗ −).

Therefore by Propositions 4.2 and 2.1,

[σ]_{KK} = χ[σ]_{Ext} = T ([φ] − [ψ]) ∈ Pext(K_{∗}(C(S^{1}) ⊗ A), K_{∗+1}(Q(K ⊗ B))).

Here we use the same notation as in the proof of Theorem 3.11. Since C(S^{1}) ⊗ A is quasidiagonal relative to B and it satisfies the UCT, we have by Corollary 4.7 that E_{σ⊕Δ̇} is a quasidiagonal set whenever Δ : C(S^{1}) ⊗ A → 𝕄(K ⊗ B) is an absorbing extension. The rest of the proof is identical with the last part of the proof of Theorem 3.11. Indeed, if η = (φ ⊕ ψ)_{∞} ⊕ δ, then the set (φ ⊕ η)(A) ∪ {u} = (Φ ⊕ δ)(A) ∪ {u} ⊂ E_{σ⊕Δ̇} is quasidiagonal and φ ⊕ η = u(ψ ⊕ η)u^{∗}. Therefore φ is approximately stably equivalent to ψ by Lemma 3.10.

■

There is a number of interesting corollaries of Theorem 5.1 where the approximate multiplicative morphisms γ_{n} implementing (iii) can be chosen to be ∗-homomorphisms. For instance this is the case when A is nuclear residually finite dimensional (γ_{n} will be finite dimensional representations) or when there is a full embedding ι : A ↪ B (γ_{n} = k(n) · ι as in Corollary 3.12).

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

        mdd@math.purdue.edu