Asymptotic unitary equivalence in KK-theory

Marius Dadarlat and Søren Eilers

June 1999

Abstract

A description of the Kasparov group KK(A, B) is given in terms of Cuntz pairs of representations and the notion of proper asymptotic unitary equivalence that we introduce here. The use of the word ‘proper’ reflects the crucial fact that all unitaries implementing the equivalence can be chosen to be compact perturbations of identity. The result has significant applications to the classification theory 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.

1 Introduction

Let A, B be C^{∗}-algebras with A separable and B σ-unital. Let 𝔼(A, B) denote the set of (A, B) C^{∗}-bimodules. The versatility of KK-theory is due to some extent to the remarkable fact, emphasized in [Ska91, Remark 10.8], that all “reasonable” equivalence relations on 𝔼(A, B), ranging from homotopy to cobordism, prove to be the same. They all give rise to the same object, the Kasparov group KK(A, B).

It is a goal of this paper to exhibit yet another equivalence relation leading to KK(A, B), see Theorem 3.8. This equivalence relation is based on the notion of proper asymptotic unitary equivalence, which we define below.

Besides its intrinsic interest, the result is motivated by applications in the classification theory of nuclear C^{∗}-algebras [DE98]. In particular it gives a better understanding of what it means that two ∗-homomorphisms φ, ψ : A → B have the same KK-theory class (see Theorems 4.2 and 4.3). It also sheds new light on the realizations of KK(A, B) for purely infinite simple nuclear C^{∗}-algebras, [Kir94], [Phi94], as it reveals a role for asymptotic unitary equivalence in a general KK-theory context. To elaborate on this point, we need the following definition. Let E be a (right) Hilbert B-module. 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

• lim_{t→∞} ∥u_{t}π(a)u_{t}^{∗} − σ(a)∥ = 0, for all a ∈ A

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

The use of the word ‘proper’ reflects the crucial fact that all unitaries are of the form “identity+compact”.

The main result is Theorem 3.8, which shows that if φ, ψ : A → M(K(H) ⊗ B) is a Cuntz pair of representations [Cun83], i.e. φ(a) − ψ(a) ∈ K(H) ⊗ B for all a ∈ A, then the class [φ, ψ] vanishes in KK(A, B) if and only if φ ⊕ γ ≊ ψ ⊕ γ for some representation γ : A → M(K(H) ⊗ B). A version of this for KK_{nuc}(A, B) [Ska88] is given in Theorem 3.10. The result is improved for K-homology. Thus if φ, ψ : A → L(H) is a Cuntz pair of faithful nondegenerate representations whose images do not contain nonzero compact operators, then [φ, ψ] = 0 in K^{0}(A) if and only if φ ≊ ψ. For a better understanding of the difference between asymptotic unitary equivalence and proper asymptotic unitary equivalence, we encourage the reader to contrast this result with Voiculescu’s Theorem 3.11.

A key step in the proof of our result was to employ the Paschke duality, its generalizations [Pas81], [Ska88] and other arguments in the same spirit, in order to produce an automorphism of φ(A) + K(H) ⊗ B which can be connected to the identity by a path continuous in the uniform topology. Then one uses a theorem of Kadison and Ringrose on automorphisms [KR67], [Ped79], by developing an idea of Lin [Lin97].

Note that our result fits nicely with the following basic example discussed in [BDF73]. Suppose that e, f ∈ L(H) are two projections with e−f ∈ K(H). The essential codimension of e in f is an integer defined as the Fredholm index of v^{∗}w, where v, u are isometries on H with vv^{∗} = e and ww^{∗} = f. This correspondence gives an isomorphism from KK(ℂ, ℂ) = K^{0}(ℂ) to ℤ. The integer associated to the pair (e, f) vanishes exactly when there is a unitary u ∈ K(H) + ℂ1 such that ueu^{∗} = f.

We are indebted to George Elliott for pointing out to us that any derivable automorphism is asymptotically inner, see Section 2.3. We are also grateful to Klaus Thomsen for suggesting substantial simplifications of our original proof of Proposition 2.15.

2 Preliminaries and background

2.1 Approximately equivalent representations

The reader is referred to [Kas80a] for an introduction to Hilbert C^{∗}-algebra modules. Throughout the paper, A is a separable C^{∗}-algebra, and all Hilbert C^{∗}-algebra modules are assumed to be countable over a σ-unital C^{∗}-algebra B. We let H_{B} denote the Hilbert module H ⊗ B, where H is an infinite dimensional, separable Hilbert space.

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 for any a ∈ A

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

We say that γ and γ′ are asymptotically unitarily equivalent, and write γ ∼_{asymp} γ′, if there exists a unitary valued norm-continuous map u : [0, ∞) → L(E′, E), u = (u_{t})_{t∈[0,∞)}, such that the map t ↦ γ(a) − u_{t}γ′(a)u_{t}^{∗} lies in C_{0}[0, ∞) ⊗ K(E) for any a ∈ A. In other words, if

(iii) lim_{t→∞} ∥γ(a) − u_{t}γ′(a)u_{t}^{∗}∥ = 0,
(iv) γ(a) − u_{t}γ′(a)u_{t}^{∗} ∈ K(E), for all t ∈ [0, ∞).

for all a ∈ A.

If σ : A → L(F) is a representation, we define σ_{∞} : A → L(F_{∞}) by σ_{∞} = σ ⊕ σ ⊕ ⋯, where F_{∞} = F ⊕ F ⊕ · · ·. Let w_{∞} : F_{∞} → F ⊕ F_{∞} be defined as w_{∞}(ξ_{1}, ξ_{2}, ξ_{3}, …) = ξ_{1} ⊕ (ξ_{2}, ξ_{3}, …). We need the following lemma of [DE98]:

Lemma 2.2 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 all a ∈ A, then σ(a) ⊕ π(a) − uπ(a)u^{∗} ∈ K(F ⊕ E), for all a ∈ A.

Lemma 2.3 Let π : A → L(E) and σ : A → L(F) be two representations. Suppose that there is a sequence of isometries v_{i} : F_{∞} → E such that

v_{i}σ_{∞}(a) − π(a)v_{i} ∈ K(F_{∞}, E),  ∥v_{i}σ_{∞}(a) − π(a)v_{i}∥ ⟶ 0,  a ∈ A

and v_{i}^{∗}v_{j} = 0 for i ≠ j. Then π ⊕ σ ∼_{asymp} π.

Proof: Extend the sequence (v_{i}) to a continuous family of isometries v : [0, ∞) → L(F_{∞}, E) by defining

v_{i+t} = (1 − t)^{1/2}v_{i} + t^{1/2}v_{i+1}, 0 ≤ t ≤ 1.

Then t ↦ v_{t}σ_{∞}(a) − π(a)v_{t} defines a function in C_{0}[0, ∞) ⊗ K(F_{∞}, E) for all a ∈ A. Now Lemma 2.2 yields a norm continuous family of unitaries u = (u_{t})_{t∈[0,∞)} such that

π(a) ⊕ σ(a) − uπ(a)u^{∗} ∈ C_{0}[0, ∞) ⊗ K(E ⊕ F).
■

Lemma 2.4 Let π : A → L(E) and σ : A → L(F) be two representations. If σ ⊕ π ∼ π, then σ ⊕ π_{∞} ∼_{asymp} π_{∞}.

Proof: If σ ⊕ π ∼ π, then σ_{∞} ⊕ π_{∞} ∼ π_{∞}. Using that E_{∞} = (E_{∞})_{∞} we may think of the unitaries given by (i)–(ii) of Definition 2.1 as isometries v′_{i} : F_{∞} ⊕E_{∞} → E_{∞} with (v′_{i})^{∗}v′_{j} = 0 for i ≠ j and

v′_{i}(σ_{∞} ⊕ π_{∞})(a) − π_{∞}(a)v′_{i} ∈ K(F_{∞}, E), ∥v′_{i}(σ_{∞} ⊕ π_{∞})(a) − π_{∞}(a)v′_{i}∥ ⟶ 0

for a ∈ A. Letting W : F_{∞} → F_{∞} ⊕ E_{∞} be the canonical isometry, we note that [σ_{∞} ⊕ π_{∞}]W = W σ_{∞}, and consequently, that π_{∞} and σ satisfy the assumptions of Lemma 2.3 using v_{i} = v′_{i}W. It follows that π_{∞} ⊕ σ ∼_{asymp} π_{∞}. ■

2.2 Absorbing representations

Definition 2.5 A representation π : A → L(E) is called absorbing if π ⊕ σ ∼ π for any representation σ : A → L(F). If A is unital, then a representation π : A → L(E) is called unitally absorbing if π ⊕ σ ∼ π for any unital representation σ : A → L(F).

A recent result of Thomsen ([Tho99, 2]) ensures the existence of absorbing representations from A to M(K(H) ⊗ B) under the modest assumptions that A and B are separable C^{∗}- algebras. If A is unital, the map can be chosen unital and unitally absorbing.

Definition 2.6 ([Ska88, 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}}(ℂ), φ_{λ} : M_{n_{λ}}(ℂ) → 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. Equivalently, for any v ∈ K(E) the map A → K(E), defined by a ↦ v^{∗}π(a)v is nuclear (see [Ska88]).

Remark 2.7 Any scalar representation θ : A → L(H) ⊂ L(H_{B}) is strictly nuclear. If either A or B is nuclear, then any completely positive contraction π : A ⟶ L(E) is strictly nuclear by [Ska88, 1.7].

Definition 2.8 A representation π : A → L(E) is called nuclearly absorbing if π ⊕ σ ∼ π for any strictly nuclear representation σ : A → L(F). If A is unital, then a representation π : A → L(E) is called unitally nuclearly absorbing if π ⊕ σ ∼ π for any unital strictly nuclear representation σ : A → L(F).

Note that a unital representation γ : A → M(K(H) ⊗ B) cannot absorb a nonunital representation.

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}. Also let γ̂ = 0 ⊕ γ : A → M(K(H ⊕ H) ⊗ B), where the zero summand acts on H ⊗ B. The following was proved in [DE98]:

Lemma 2.9 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 = 0 ⊕ γ is a nonunital nuclearly absorbing representation.

Note that one has a version of Lemma 2.9 for absorbing representations.

Definition 2.10 Let A be a separable C^{∗}-algebra. A faithful scalar representation of infinite multiplicity 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.

Proposition 2.11 Let A be a unital separable C^{∗}-algebra. If θ : A ⟶ L(H_{B}) is a unital faithful scalar representation of infinite multiplicity, then θ is unitally nuclearly absorbing.

Proof: This follows from the arguments of [Kas80a], as pointed out in [Ska88]. Details can be found in [DE98] ■

2.3 Automorphisms and derivations

Let A denote a unital separable C^{∗}-algebra, and denote as in [Ped79] by B(A) the Banach algebra of linear operators on A and by Aut(A) and Der(A) the subsets of B(A) consisting of ∗-automorphisms and ∗-derivations, respectively. Unless otherwise specified, we are going to consider B(A) and these subsets as equipped with the uniform topology. Thus Aut_{0}(A) refers to the connected component of the identity of Aut(A) in the uniform topology. Note that analytic functional calculus is at our disposal in B(A), and that δ ↦ exp δ is a continuous map from Der(A) to Aut(A).

We can define inner automorphisms and derivations by

Ad(u)(a) = uau^{∗} ad(ih)(a) = i[h, a] = i(ha − ah)

for u, h ∈ A which are unitary and selfadjoint, respectively. We get exp ad(ih) = Ad(exp ih) in this setting. We call elements of Aut(A) or Der(A) asymptotically inner when they are the pointwise limits of uniformly continuous one-parameter families of inner automorphisms or derivations. Thus an asymptotically inner automorphism α is one for which there is a continuous path of unitaries (u_{t})_{t∈[0,∞)} such that

lim_{t→∞} ∥Ad(u_{t})(a) − α(a)∥ = 0, a ∈ A.

Similarly, a derivation δ is asymptotically inner if there is a norm continuous bounded family of selfadjoint elements (h_{t})_{t∈[0,∞)} such that

lim_{t→∞} ∥ad(ih_{t})(a) − δ(a)∥ = 0, a ∈ A. (1)

Lemma 2.12 If A is a unital separable C^{∗}-algebra, then any derivation δ ∈ Der(A) is asymptotically inner. In fact, one may choose h_{t} in (1) with ∥h_{t}∥ ≤ ∥δ∥.

Proof: By [Ped79, 8.6.12] there is a sequence h_{n} of selfadjoint elements of A such that lim_{n→∞} ∥ad(ih_{n})(a) − δ(a)∥ = 0 for all a ∈ A. We may choose h_{n} such that ∥h_{n}∥ ≤ ∥δ∥. By linear interpolation, we find a norm continuous bounded family of selfadjoint elements (h_{t})_{t∈[0,∞)} satisfying (1). ■

The proof of the following elementary lemma is left to the reader.

Lemma 2.13 The map exp : Der(A) → Aut(A) is continuous in the point-norm topology. That is, if lim_{n→∞} ∥δ_{n}(a)−δ(a)∥ = 0 for all a ∈ A, then lim_{n→∞} ∥ exp(δ_{n})(a)−exp(δ)(a)∥ = 0, for all a ∈ A.

Lemma 2.14 If A is a unital separable C^{∗}-algebra, then any automorphism in Aut_{0}(A) is asymptotically inner. The one-parameter family can be found within Aut_{0}(A) ∩ Inn(A).

Proof: If α ∈ Aut_{0}(A), then by [Ped79, 8.7.8] α is a product of finitely many automorphisms of the form exp δ, δ ∈ Der(A). Thus it suffices to assume that in fact α = exp δ. Using Lemma 2.13, the result now follows from exponentiation of the path constructed in Lemma 2.12, noting that exp(Der(A)) ⊆ Aut_{0}(A). ■

Proposition 2.15 Let A be a unital separable C^{∗}-algebra. If (α_{t})_{t∈[0;∞)} is a uniformly continuous family in Aut(A) with α_{0} = id_{A}, then there exists a continuous family (v_{t})_{t∈[0;∞)} of unitaries in A with v_{0} = 1 such that

lim_{t⟶∞} ∥α_{t}(a) − Ad(v_{t})(a)∥ = 0 a ∈ A.

Proof: Applying uniform continuity separately on each interval [n, n + 1] we see that we can divide [0, ∞) into pieces by 0 = t_{0} < t_{1} < ⋯ < t_{j} < t_{j+1} < ⋯ with t_{j} ⟶ ∞ such that when we set

μ^{(j)} = sup_{t∈[t_{j−1},t_{j}]} ∥α_{t} − α_{t_{j}} ∥,

μ^{(j)} < 2 and μ^{(j)} ⟶ 0 for j ⟶ ∞.

For any j ≥ 1, let β^{(j)} = α_{t_{j}} ∘ α_{t_{j−1}}^{−1} and note that ∥id −β^{(j)}∥ ≤ μ^{(j)}. By [Ped79, 8.7.7], when we set

δ^{(j)} = Log(β^{(j)})

in B(A), we get a sequence of ∗-derivations. Furthermore, ∥δ^{(j)}∥ ⟶ ∥Log(id)∥ = 0. Using

Lemma 2.12 we choose selfadjoint families (h_{t}^{(j)})_{t∈[0,∞)} such that ∥h_{t}^{(j)}∥ ≤ ∥δ^{(j)}∥ for all t, j,

h_{0}^{(j)} = 0 and

∥δ^{(j)}(a) − ad(ih_{t}^{(j)})(a)∥ ⟶ 0 (2)

for all a. Note that ∥e^{ih_{t}^{(j)}} − 1∥ ≤ e^{∥h_{t}^{(j)}∥} − 1 ≤ e^{∥^{δ^{(j)}}∥} − 1.

Fix a dense sequence (a_{i})_{i=0}^{∞} of the unit ball of A. When k ≥ j ≥ 1 we set

F_{k}^{(j)} = {β^{(j−1)} ∘ ⋯ ∘ β^{(1)}(a_{i}) | i ≤ k},

interpreted in the obvious way when j = 1. Exponentiating each family h_{t}^{(j)} and then reparametrizing using a homeomorphism between [t_{j}, ∞) and [0, ∞), we get continuous families of unitaries v_{t}^{(j)} with the properties

v_{t}^{(j)} = 1 t ∈ [0, t_{j}]
∥1 − v_{t}^{(j)}∥ ≤ e^{∥^{δ^{(j)}}∥} − 1 t ∈ [t_{j}, t_{j+1}]
∥β^{(j)}(a) − Ad(v_{t}^{(j)})(a)∥ ≤ 2^{−j−k} t ∈ [t_{k}, t_{k+1}], a ∈ F_{k}^{(j)}, k > j

This is arranged using (2), noting that the condition on [t_{j}, t_{j+1}] in each case holds true on any element of the form exp(ih_{t}^{(j)}).

When t ∈ [t_{j}, t_{j+1}] and i ≤ j we may use our assumption on v_{t}^{(j−1)} on the interval [t_{k}, t_{k+1}] with k = j to see that

∥α_{t_{j−1}}(a_{i}) − Ad(v_{t}^{(j−1)} ⋯ v_{t}^{(1)})(a_{i})∥
≤ ∥β^{(j−1)}[β^{(j−2)} ∘ ⋯ ∘ β^{(1)}(a_{i})] − Ad(v_{t}^{(j−1)})[β^{(j−2)} ∘ ⋯ ∘ β^{(1)}(a_{i})]∥
+ ∥β^{(j−2)} ∘ ⋯ ∘ β^{(1)}(a_{i}) − Ad(v_{t}^{(j−2)} ⋯ v_{t}^{(1)})(a_{i})∥
≤ 2^{−2j+1} + ∥α_{t_{j−2}}(a_{i}) − Ad(v_{t}^{(j−2)} ⋯ v_{t}^{(1)})(a_{i})∥

We can iterate this argument to prove that under these circumstances,

∥α_{t_{j−1}}(a_{i}) − Ad(v_{t}^{(j−1)} ⋯ v_{t}^{(1)})(a_{i})∥ ≤ 2^{−2j+1} + ⋯ + 2^{−j−1} < 2^{−j} (3)

We may now define

u_{t} = lim_{j⟶∞} v_{t}^{(j)} ⋯ v_{t}^{(1)}

since for every t, the sequence is eventually constant, as v_{1}^{(j)} = 1 on the interval [0, t_{j}]. This is then obviously a continuous family of unitaries. Fix now i and consider a_{i}. When j ≥ i and t ∈ [t_{j}, t_{j+1}] we have by (3), the definition of μ^{(j)}, and the condition of v_{t}^{(j)} on [t_{j}, t_{j+1}] that

∥α_{t}(a_{i}) − Ad(u_{t})(a_{i})∥ = ∥α_{t}(a_{i}) − Ad(v_{t}^{(j)} ⋯ v_{t}^{(1)})(a_{i})∥
≤ μ^{(j)} + 2μ^{(j+1)} + ∥α_{t_{j−1}}(a_{i}) − Ad(v_{t}^{(j)} ⋯ v_{t}^{(1)})(a_{i})∥
≤ μ^{(j)} + 2μ^{(j+1)} + 2∥v_{t}^{(j)} − 1∥ + ∥α_{t_{j−1}}(a_{i}) − Ad(v_{t}^{(j−1)} ⋯ v_{t}^{(1)})(a_{i})∥
≤ μ^{(j)} + 2μ^{(j+1)} + 2(e^{∥^{δ^{(j)}}∥} − 1) + 2^{−j},

showing that the norm does converge to zero as t ⟶ ∞. ■

3 The main result

In all of Section 3, we only work with infinite dimensional, separable Hilbert spaces, so all Hilbert spaces in this paper are isomorphic. However, we introduce the following notation to aid the reader in distinguishing between different instances of them. We start with a separable Hilbert space H_{1} and define

H_{m} = \overbrace{H_{1} ⊕ ⋯ ⊕ H_{1}}^{m} .

for any m ∈ ℕ. There are now canonical identifications between, say, M_{2}(K(H_{1}) ⊗ B) and K(H_{2})⊗B, and we shall employ them tacitly in the following. However, sometimes we choose not to apply the (non-canonical) isomorphisms between, e.g., K(H_{1}) ⊗ B and K(H_{2}) ⊗ B, as we feel this helps to clarify our constructions.

We work with the multiplier algebras M(K(H_{m}) ⊗ B), as well as the corona algebras

Q(K(H_{m}) ⊗ B) = M(K(H_{m}) ⊗ B)/K(H_{m}) ⊗ B.

The quotient map from M(K(H_{m}) ⊗ B) to Q(K(H_{m}) ⊗ B) is denoted by π_{m}, and whenever there is need for distinction, we write 1_{m} and 0_{m} for the identity and zero elements of these algebras.

3.1 KK-theory

We depend on Kasparov and Skandalis’ bifunctors KK and KK_{nuc} in the paper, and refer the reader to [Kas80a] and [Ska88] for their definition. We recall that throughout the paper A is a separable C^{∗}-algebra and B is a σ-unital C^{∗}-algebra. All Hilbert modules E are countable. If H_{B} = H ⊗ B, then L(H_{B}) ≅ M(K(H) ⊗ B). We need both the Fredholm picture and the Cuntz picture of KK(A, B). In the Fredholm picture (see [Hig87, 2.1]), KK(A, B) is described in terms of triples (φ_{0}, φ_{1}, u), which we call KK-cycles or just cycles, where φ_{i} : A ⟶ L(E_{i}) are ∗-homomorphisms and u ∈ L(E_{0}, E_{1}) satisfies

uφ_{0}(a) − φ_{1}(a)u ∈ K(E_{0}, E_{1}) (4) φ_{0}(a)(u^{∗}u − 1) ∈ K(E_{0}), φ_{1}(a)(uu^{∗} − 1) ∈ K(E_{1}). (5)

The set of all cycles as above is denoted by 𝔼(A, B). A cycle is degenerated if

uφ_{0}(a) − φ_{1}(a)u = 0 φ_{0}(a)(u^{∗}u − 1) = 0 φ_{1}(a)(uu^{∗} − 1) = 0, a ∈ A.

An operatorial homotopy through KK-cycles is a homotopy (φ_{0}, φ_{1}, u_{t}), where the map t ↦ u_{t} is norm continuous. A theorem of Kasparov [Kas80b] shows that KK(A, B) is isomorphic to the quotient of 𝔼(A, B) by the equivalence relation generated by addition of degenerated cycles, unitary equivalence and operatorial homotopy.

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]). The quotient of 𝔼_{h}(A, B) by homotopy equivalence is a group KK_{h}(A, B) isomorphic to KK(A, B). The isomorphism

KK_{h}(A, B) ⟶ KK(A, B) (6)

maps [φ, ψ] to [φ, ψ, 1]. A quick way to see that the map (6) is surjective is to show that any cycle is equivalent to a cycle where u is a unitary. Then since (φ_{0}, φ_{1}, u) is unitarily equivalent to (uφ_{0}u^{∗}, φ_{1}, 1), we see that (6) maps [uφ_{0}u^{∗}, φ_{1}] to [φ_{0}, φ_{1}, u]. The injectivity of (6) is proved by applying a similar method to an operatorial homotopy.

One defines KK_{nuc} by restricting attention to the case where φ_{i} are strictly nuclear ∗- homomorphisms. Hence KK = KK_{nuc} when A or B is nuclear. Similarly KK_{nuc,h}(A, B) is defined by working only with Cuntz pairs (and homotopies) consisting of strictly nuclear representations. And again one has an isomorphism

KK_{nuc,h}(A, B) ⟶ KK_{nuc}(A, B).

This isomorphism is proved using the description of KK_{nuc}(A, B) based on operatorial homotopy given by [Ska88, 2.6].

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

3.2 Proper asymptotic unitary equivalence

Lemma 3.1 If φ, ψ : A ⟶ M(K(H)⊗B) is a Cuntz pair of representations, and (u_{t})_{t∈[1,∞)} is a continuous path of unitaries in M(K(H) ⊗ B) satisfying

u_{t}φ(a)u_{t}^{∗} − ψ(a) ∈ K(H) ⊗ B,

for all t ∈ [1, ∞), then [φ, ψ] = [φ, u_{1}φu_{1}^{∗}] in KK_{h}(A, B).

Proof: If Φ_{0}, Φ_{1} : A → M(K(H) ⊗ C[0, 1]) = C_{b,strictly}([0, 1], L(H)) are defined by

Φ_{0}(a)(s) = φ(a) Φ_{1}(a)(s) = \begin{cases}Ad(u_{1/s}) ∘ φ(a) & s > 0 \\ ψ(a) & s = 0 \end{cases}

then (Φ_{0}, Φ_{1}) is a homotopy of Cuntz pairs from (φ, ψ) to (φ, u_{1}φu_{1}^{∗}). ■

Definition 3.2 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, ∞).

Lemma 3.3 If φ ≊ ψ, then φ(a) − ψ(a) ∈ K(H) ⊗ B for all a ∈ A and [φ, ψ, 1] = 0 in KK(A, B).

Proof: Let (u_{t})_{t∈[0,∞)} be as in Definition 3.2. The first part of the lemma is obvious since u_{t} ∈ K(H) ⊗ B + ℂ1. Then [φ, ψ, 1] = [φ, u_{1}φu_{1}^{∗}, 1] by the lemma above. Since (φ, u_{1}φu_{1}^{∗}, 1) is unitarily equivalent to (φ, φ, u_{1}^{∗}) and u_{1}^{∗} ∈ K(H) ⊗ B + ℂ1, we obtain that [φ, ψ, 1] = [φ, φ, 1] = 0. ■

Lemma 3.4 Let φ, ψ, γ, σ : A → M(K(H)⊗B) be unital representations. If φ⊕γ ≊ ψ ⊕γ, and γ ∼_{asymp} σ, then φ ⊕ σ ≊ ψ ⊕ σ.

Proof: By assumption, there is a continuous path of unitaries u : [0, ∞) → U(K(H_{2}) ⊗ B + ℂ1_{2}) such that

u_{t}(φ(a) ⊕ γ(a))u_{t}^{∗} − ψ(a) ⊕ γ(a) ∈ K(H_{2}) ⊗ B,

lim_{t→∞} ∥u_{t}(φ(a) ⊕ γ(a))u_{t}^{∗} − ψ(a) ⊕ γ(a)∥ = 0 (7)

for all a ∈ F_{n}. Since γ ∼_{asymp} σ, there is a continuous path of unitaries (v_{t})_{t∈[0,∞)} such that

v_{t}γ(a)v_{t}^{∗} − σ(a) ∈ K(H) ⊗ B, lim_{t→∞} ∥v_{t}γ(a)v_{t}^{∗} − σ(a)∥ = 0 (8)

for all a ∈ F_{n}. Clearly w_{t} = (1 ⊕ v_{t})u_{t}(1 ⊕ v_{t}^{∗}) is a unitary in K(H_{2}) ⊗ B + ℂ1_{2}. If a ∈ A, then from (7) and (8)

\begin{matrix}∥w_{t}(φ(a) ⊕ σ(a))w_{t}^{∗} − ψ(a) ⊕ σ(a)∥ = ∥u_{t}(φ(a) ⊕ v_{t}^{∗}σ(a)v_{t})u_{t}^{∗} − ψ(a) ⊕ v_{t}^{∗}σ(a)v_{t}∥ ≤ \\ 2∥v_{t}^{∗}σ(a)v_{t} − γ(a)∥ + ∥u_{t}(φ(a) ⊕ γ(a))u_{t}^{∗} − φ(a) ⊕ γ(a)∥ → 0 \end{matrix}

By passing to the generalized Calkin algebra in the above inequalities we also obtain

w_{t}(φ(a) ⊕ σ(a))w_{t}^{∗} − ψ(a) ⊕ σ(a) ∈ K(H_{2}) ⊗ B, a ∈ A.
■

If Φ : A → M(K(H) ⊗ B) is a representation, we define a C^{∗}-algebra D_{Φ} by

D_{Φ} = {b ∈ M(K(H) ⊗ B) | [b, Φ(A)] ⊂ K(H) ⊗ B}.

We use the dot as a shorthand to indicate composition by the quotient map π : M(K(H) ⊗ B) ⟶ Q(K(H) ⊗ B). Thus, Φ̇ = π ∘ Φ maps from A to Q(K(H) ⊗ B). Furthermore, if X ⊆ Q(K(H) ⊗ B), we denote by X^{c} the commutator of X in Q(K(H) ⊗ B). One checks directly that

\xymatrix{0 \ar[r] & K(H) ⊗ B \ar@{^{(}->}[r]^{j} & D_{Φ} \ar[r]^{π} & Φ̇(A)^{c} \ar[r] & 0} (9)

is a short exact sequence of C^{∗}-algebras.

Lemma 3.5 Let A be a unital separable C^{∗}-algebra and let B be a σ-unital C^{∗}-algebra. Let Φ : A → M(K(H) ⊗ B) be a unital representation and let w_{i} be unitaries in D_{Φ}. Suppose that [Φ, Φ, w_{1}] = [Φ, Φ, w_{2}] in KK(A, B). Then there is a unital representation γ : A → M(K(H) ⊗ B) such that

(Φ ⊕ γ, Φ ⊕ γ, w_{1} ⊕ 1) ∼_{oh} (Φ ⊕ γ, Φ ⊕ γ, w_{2} ⊕ 1) (10)

If Φ is strictly nuclear and [Φ, Φ, w_{1}] = [Φ, Φ, w_{2}] in KK_{nuc}(A, B), then γ can be chosen to be strictly nuclear.

Proof: By [Kas80b, §6, Theorem 1], there is a degenerate cycle (γ_{0}, γ_{1}, v) with γ_{i} : A → L(E_{i}) and v ∈ L(E_{0}, E_{1}) such that

(Φ ⊕ γ_{0}, Φ ⊕ γ_{1}, w_{1} ⊕ v) ∼_{oh} (Φ ⊕ γ_{0}, Φ ⊕ γ_{1}, w_{1} ⊕ v) (11)

Note that if Φ is strictly nuclear, then γ can be chosen to be strictly nuclear by [Ska88, Proposition 2.2(c)]. Let W_{t} be the continuous path from w_{1} ⊕ v to w_{2} ⊕ v which implements (11).

Since (γ_{0}, γ_{1}, v) is degenerated, we have that

vγ_{0}(a) − γ_{1}(a)v = 0, γ_{0}(a)(1 − v^{∗}v) = 0, γ_{1}(a)(1 − vv^{∗}) = 0, a ∈ A.

Let p_{i} = γ_{i}(1). After replacing E_{i} by p_{i}E_{i}, v by p_{1}vp_{0} and W_{t} by (1 ⊕ p_{1})W_{t}(1 ⊕ p_{0}) we may assume that (11) holds with γ_{i} unital and v a unitary.

Next we observe that (Φ ⊕ γ_{0}, Φ ⊕ γ_{0}, (1 ⊕ v^{∗})W_{t}) is a KK-cycle for all t, and we see from (11) that this implements an operatorial homotopy

(Φ ⊕ γ_{0}, Φ ⊕ γ_{0}, w_{1} ⊕ 1) ∼_{oh} (Φ ⊕ γ_{0}, Φ ⊕ γ_{0}, w_{2} ⊕ 1) (12)

To complete the proof we need to arrange that E_{i} = H_{B}. Let θ : A → L(H_{B}) be a unital faithful scalar representation of infinite multiplicity. Let (θ, θ, 1) be the corresponding trivial triple. From (12), we have that

(Φ ⊕ γ_{0} ⊕ θ, Φ ⊕ γ_{0} ⊕ θ, w_{1} ⊕ 1 ⊕ 1) ∼_{oh} (Φ ⊕ γ_{0} ⊕ θ, Φ ⊕ γ_{0} ⊕ θ, w_{2} ⊕ 1 ⊕ 1) (13)

Let z : E_{0} ⊕ H_{B} → H_{B} be a unitary given by Kasparov’s stabilization theorem. Setting γ = z(γ_{0} ⊕ θ)z^{∗} we see that (10) is a consequence of (13). Note that γ is strictly nuclear if γ_{0} is so. ■

Proposition 3.6 Let A be a separable C^{∗}-algebra and let B be a σ-unital C^{∗}-algebra. Let φ, ψ : A → M(K(H) ⊗ B) be representations such that φ(a) − ψ(a) ∈ K(H) ⊗ B for all a ∈ A. Then if [φ, ψ, 1] = 0 in KK(A, B), there is a representation γ : A → M(K(H) ⊗ B) such that φ ⊕ γ ≊ ψ ⊕ γ.

When

(a) A has a unit and φ, ψ are unital,

we may arrange that γ is unital. When

(b) φ and ψ are strictly nuclear, and [φ, ψ, 1] = 0 in KK_{nuc}(A, B)

we may arrange that γ is strictly nuclear. When both (a) and (b) hold, we may arrange that γ is unital and strictly nuclear.

Proof: Part 1: We first prove the existence of γ under the unitality condition (a).

Step 1a: Let σ = φ_{∞} ⊕ ψ_{∞} and set Φ′ = φ ⊕ σ and Ψ′ = ψ ⊕ σ. We note that then Ψ′ = Ad(u′) ∘ Φ′ for some unitary u′ ∈ L(H_{B}). We have

[Φ′, Φ′, u′] = [Φ′, Ψ′, 1] = [φ, ψ, 1] = 0 = [Φ′, Φ′, 1] (14)

in KK(A, B), so by Lemma 3.5 there is a unital representation γ′ : A → L(H_{B}) such that

(Φ′ ⊕ γ′, Φ′ ⊕ γ′, u′ ⊕ 1) ∼_{oh} (Φ′ ⊕ γ′, Φ′ ⊕ γ′, 1 ⊕ 1).

Without loss of generality we can assume that γ̇′ : A ⟶ L(H_{B})/K(H_{B}) is injective.

Step 1b: Define Φ = Φ′ ⊕ γ′, Ψ = Ψ′ ⊕ γ′, u = u′ ⊕ 1. Then Ψ = Ad(u) ∘ Φ, and (Φ, Φ, u) ∼_{oh} (Φ, Φ, 1). Using this operatorial homotopy, we show that u is homotopic in U(D_{Φ}) to a unitary in K(H) ⊗ B + ℂ1. Indeed, if ω_{s} ∈ M(K(H) ⊗ B), s ∈ [0, 1] is a norm continuous path of operators, implementing the operatorial homotopy, with ω_{0} = u, ω_{1} = 1 then

[Φ(a), ω_{s}] ∈ K(H) ⊗ B
Φ(a)(ω_{s}ω_{s}^{∗} − 1), Φ(a)(ω_{s}^{∗}ω_{s} − 1) ∈ K(H) ⊗ B

for all s ∈ [0, 1]. Since Φ(1) = 1, we see that ω̇_{s} is a continuous path of unitaries in Φ̇(A)^{c} joining u̇ with 1. Since U(D_{Φ}) → U(Φ̇(A)^{c}) is a fibration with fiber U(K(H ⊗ B) + ℂ1), we obtain that u is homotopic to an element of U(K(H ⊗ B) + ℂ1).

Step 1c: We prove that there is a continuous path of unitaries (v_{t})_{t∈[0,∞)} with

v_{t} ∈ E_{Φ} =^{def} Φ(A) + K(H) ⊗ B,

such that

lim_{t→∞} ∥v_{t}Φ(a)v_{t}^{∗} − Ψ(a)∥ = 0, a ∈ A. (15)

We have seen that here is a unitary w ∈ U(K(H) ⊗ B + ℂ1), which is homotopic to u in the unitary group of D_{Φ}. Since Ad(u) is homotopic to Ad(w) in the space Aut(E_{Φ}) endowed with the uniform topology, it follows that Ad(w^{∗}u) ∈ Aut_{0}(E_{Φ}). Let (α_{s})_{s∈[0,1]} be a uniformly continuous path in Aut(E_{Φ}) connecting id to α = Ad(w^{∗}u). Let D be the unital C^{∗}-subalgebra of E_{Φ} generated by all subalgebras of the form

α_{s_{1}}^{j_{1}} ∘ α_{s_{2}}^{j_{2}} ∘ ⋯ ∘ α_{s_{n}}^{j_{n}} ∘ Φ(A)

where n ≥ 1, j_{k} ∈ ℤ, and s_{k} ∈ [0, 1] ∩ ℚ. Then D is separable, since A is separable, and α_{t}(D) = D for all t ∈ [0, 1]. This proves that Ad(w^{∗}u) ∈ Aut_{0}(D). It follows by Proposition 2.14 that Ad(w^{∗}u) is asymptotically inner in Aut(D). Thus there is a continuous path of unitaries (v_{t})_{t∈[0,∞)} ⊂ wD ⊂ E_{Φ} with v_{0} = w and

lim_{t→∞} ∥v_{t}dv_{t}^{∗} − udu^{∗}∥ = 0, d ∈ D.

Therefore

(16)  lim_{t→∞} ∥v_{t}Φ(a)v_{t}^{∗} − Ψ(a)∥ = lim_{t→∞} ∥v_{t}Φ(a)v_{t}^{∗} − uΦ(a)u^{∗}∥ = 0

for all a ∈ A. This proves (15).

Step 1d: Let γ = σ ⊕ γ′. Since Φ = φ ⊕ γ and Ψ = ψ ⊕ γ, in order to complete the proof that γ has the desired properties in the presence of (a) it suffices to show that there is a continuous path of unitaries u_{t} ∈ K(H) ⊗ B + ℂ1 such that

lim_{t→∞} ∥u_{t}Φ(a)u_{t}^{∗} − Ψ(a)∥ = 0, a ∈ A. (17)

By definition of E_{Φ}, there are x_{t} ∈ A and y_{t} ∈ K(H) ⊗ B such that v_{t} = Φ(x_{t}) + y_{t}. Since we have arranged that Φ̇ is a unital monomorphism, (x_{t}) must be a continuous path of unitaries. Moreover

lim_{t→∞} ∥x_{t}ax_{t}^{∗} − a∥ = 0, a ∈ A (18)

as a consequence of (15), since Φ̇ = Ψ̇ is norm-preserving. Define

u_{t} = v_{t}Φ(x_{t})^{∗} = 1 + y_{t}Φ(x_{t})^{∗}.

Then, from (15) and (18),

∥u_{t}Φ(a)u_{t}^{∗} − Ψ(a)∥ ≤ ∥v_{t}Φ(a)v_{t}^{∗} − Ψ(a)∥ + ∥v_{t}Φ(x_{t}^{∗}ax_{t} − a)v_{t}^{∗}∥ → 0, as t → ∞, a ∈ A.

This proves (17) and concludes the proof of the unital case of the proposition.

Part 2: We now briefly indicate how to choose γ strictly nuclear in the presence of both (a) and (b). Note that in Step 1a if φ, ψ are strictly nuclear, then so is σ. Also, in this case one gets that then (14) holds in KK_{nuc}(A, B). We then may choose γ′ strictly nuclear in Step 1c by appealing to the nuclear version of Lemma 3.5 above. The remaining steps are carried out verbatim.

Part 3: The general claims are straightforward consequences of the unital cases. Indeed, after replacing A by Ã and φ, φ, γ by φ̃, φ̃, γ̃, we may assume that the C^{∗}-algebra A and all these maps are unital. When (b) holds, we still have [φ̃, φ̃, 1] = 0 in KK_{nuc}(Ã, B̃) and that γ̃ is unital and strictly nuclear. ■

Notes 3.7 The idea of invoking results about the automorphisms of C^{∗}-algebras of the form E_{Φ} originates with Lin ([Lin97]) in the case Φ = d_{ι} with d_{ι} defined as in Section 4 below.

Because of Lemma 3.3, the proposition above gives a new characterization of equivalence of cycles in the Cuntz picture of KK-theory:

Theorem 3.8 Let A be a (unital) separable C^{∗}-algebra and let B be a separable C^{∗}-algebra. Let φ, ψ : A → M(K(H)⊗B) be (unital) representations such that φ(a)−ψ(a) ∈ K(H)⊗B for all a ∈ A. Then the following are equivalent:

(i) [φ, ψ, 1] = 0 in KK(A, B) (ii) [φ, ψ] = 0 in KK_{h}(A, B) (iii) There exists a (unital) representation σ : A → M(K(H) ⊗ B) with φ ⊕ σ ≊ ψ ⊕ σ (iv) For any (unital) absorbing representation γ : A → M(K(H)⊗B), φ⊕γ_{∞} ≊ ψ ⊕γ_{∞}

Proof: We saw that (i)⟺(ii) in Section 3.1. We have proved (ii)⟹(iii) in the proposition above, and (iii)⟹(ii) follows by Lemma 3.3. When (iii) holds, we have φ ⊕ σ ⊕ γ_{∞} ≊ ψ ⊕ σ ⊕ γ_{∞}. In view of Lemma 3.4, this implies φ ⊕ γ_{∞} ≊ ψ ⊕ γ_{∞} since σ ⊕ γ_{∞} ∼_{asymp} γ_{∞} by Proposition 2.4 and the absorption property of γ. This proves (iii)⟹(iv). Finally, to prove (iv)⟹(iii), we note that an absorbing representation does exist by [Tho99, 2.4,2.7]. ■

Remark 3.9 It is clear from the proof of Theorem 3.8, that the implications (i)⟺(ii)⟹(iii) hold for any σ-unital C^{∗}-algebra B, while (iii)⟹(i) holds whenever absorbing representations γ : A → M(K(H) ⊗ B) do exist, such as in the case when B is separable.

The similar characterization in the setting of KK_{nuc} is valid for any σ-unital B:

Theorem 3.10 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 (unital) strictly nuclear representations such that φ(a) − ψ(a) ∈ K(H) ⊗ B for all a ∈ A. Then the following are equivalent:

(i) [φ, ψ, 1] = 0 in KK_{nuc}(A, B) (ii) [φ, ψ] = 0 in KK_{nuc,h}(A, B) (iii) There exists a (unital) strictly nuclear representation σ : A → M(K(H) ⊗ B) with φ ⊕ σ ≊ ψ ⊕ σ (iv) For any (unital) nuclearly absorbing and strictly nuclear representation γ : A → M(K(H) ⊗ B), φ ⊕ γ_{∞} ≊ ψ ⊕ γ_{∞}

Proof: Arguments parallel to those in the proof above show that (i)–(iv) are equivalent; in the latter case because γ absorbs σ by definition. Furthermore, (iii)⟹(iv) is clear. To prove (iv)⟹(iii), we note that a strictly nuclear and nuclearly absorbing representation does exist, cf. Proposition 2.11. ■

3.3 K-homology

We say that a ∗-representation φ : A ⟶ L(H) is admissible if it is faithful and nondegenerate and satisfies φ(A) ∩ K(H) = {0}. Equivalently, φ is nondegenerate and φ̇ : A → Q(H) is injective. The main result in [Voi76] states that any pair of admissible representations satisfies φ ∼ ψ. In fact, more is proved:

Theorem 3.11 [Voiculescu] If φ and ψ are admissible representations, we have φ ∼_{asymp} ψ.

Proof: The results of [Voi76] are stated only for unital A. To deal with the general case, note that if A is unital, then any admissible representation is necessarily unital. Moreover, if A is nonunital and φ is admissible, then φ̃ : Ã → L(H) is admissible. Therefore we may assume that A and both of φ and ψ are unital.

The proof of [Voi76, Theorem 1.3] gives a sequence of isometries v_{n} ∈ L(H) which satisfy

v_{n}φ_{∞}(a) − ψ(a)v_{n} ∈ K(H),  ∥v_{n}φ_{∞}(a) − ψ(a)v_{n}∥ ⟶ 0,  a ∈ A

Lemma 2.3 shows that φ ⊕ ψ ∼_{asymp} ψ, and the proof is complete by symmetry. ■

Theorem 3.12 Let A be a separable C^{∗}-algebra and let φ, ψ : A ⟶ L(H) be a Cuntz pair of admissible representations. Then [φ, ψ] = 0 in KK(A, ℂ) = K^{0}(A) if and only if φ ≊ ψ.

Proof: One implication is covered by Lemma 3.3. To prove the other, arguing as in the proof of Theorem 3.11, we may assume that A, φ and ψ are unital. By Theorem 3.11 we get a continuous family of unitaries (u_{t})_{t∈[0,∞)} with

u_{t}φ(a)u_{t}^{∗} − ψ(a) ∈ C_{0}([0, ∞)) ⊗ K(H). (19)

We assume that [φ, ψ] = 0 and conclude from Lemma 3.1 and (19) that also [φ, u_{1}φu_{1}^{∗}] = 0 in KK_{h}(A, B). Since (φ, φ, u_{1}^{∗}) is unitarily equivalent to (φ, u_{1}φu_{1}^{∗}, 1), we conclude that [φ, φ, u_{1}] = [φ, φ, u_{1}^{∗}] = 0 in KK(A, B). By Paschke duality, u̇_{1} is homotopic to 1 in the unitary group of the commutant of φ̇(A) in the Calkin algebra. Indeed, since KK(A, ℂ) ≅ K_{1}(φ̇(A)^{c}) we have that [u̇_{1}] = 0 in K_{1}(φ̇(A)^{c}) and the latter group is isomorphic to the quotient the unitary group of φ̇(A)^{c} by the path component of the identity by [Pas81, Lemma 3].

Then as in Step 1b of the proof of Proposition 3.6, using the principal fibration

U(K(H) + ℂ1) → U(D_{φ}) → U(φ̇(A)^{c})

we conclude that u_{1} is homotopic to 1 in D_{φ}. Note that any element v of such a homotopy satisfies vφ(a)v^{∗} − ψ(a) ∈ K(H). We may (and shall) hence assume that in fact u_{0} = 1 in (19).

We define (α_{t})_{t∈[0,∞)} in Aut_{0}(E_{φ}) by α_{t} = Ad(u_{t}), noting that this is a uniformly continuous family there. Hence Proposition 2.15 applies to give us a continuous family (v_{t})_{t∈[0,∞)} such that

lim_{t⟶∞} ∥α_{t}(x) − Ad(v_{t})(x)∥ = 0

for any x ∈ E_{φ}. Combining this with (19) we get that v_{t} satisfies

lim_{t⟶∞} ∥v_{t}φ(a)v_{t}^{∗} − ψ(a)∥ = 0

for any a ∈ A. Since φ̇ is faithful, we can replace (v_{t})_{t∈[0,∞)} by a family of unitaries in K(H) + ℂ1 as in Step 1d of the proof of Proposition 3.6. ■

4 Applications

Lin [Lin97] showed that if ι : A → B is a unital embedding, if A is nuclear, and either A or B is simple, then the map d_{ι} : A ⟶ M(K(H) ⊗ B) defined by d_{ι}(a) = 1⊗ι(a) is absorbing. This was generalized in [DE98] as follows.

Theorem 4.1 Let A be a unital separable C^{∗}-algebra and let ι : A ↪ B be a unital full embedding, i.e. a unital embedding with the property that span Bι(a)B is dense in B for all nonzero a ∈ A. Then d_{ι} is nuclearly absorbing.

The following two theorems are immediate applications of Theorem 3.10. Their proofs are given in full in [DE98], however we indicate here the basic idea behind them. Theorem 4.2 generalizes a result of Lin [Lin97], where A was assumed to be nuclear and to satisfy the universal coefficient theorem of Rosenberg and Schochet, and either A or B were simple.

Theorem 4.2 Let A be a unital, separable C^{∗}-algebra and let B be a unital C^{∗}-algebra. Assume that there exists a unital full embedding ι : A → B. Suppose that φ, ψ : A ⟶ B are two nuclear unital ∗-homomorphisms with [φ] = [ψ] in KK_{nuc}(A, B). Then there is a sequence of unitaries u ∈ U_{n+1}(B) such that

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

Proof: We give only the idea of the proof. Since φ and ψ assume only compact values, we have

[φ, ψ, 1] = [φ, ψ, 0] = [φ, 0, 0] − [ψ, 0, 0] = 0

in KK_{nuc}(A, B). The representation d_{ι} is nuclearly absorbing by Theorem 4.1. Therefore φ ⊕ d_{ι} ≊ ψ ⊕ d_{ι} by Theorem 3.10. Compressing this by p_{n} = (n + 1)·ι(1), yields the result in the statement, since p_{n} is an approximate unit of projections of K(H) ⊗ B which commutes with d_{ι}. ■

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. The following result generalizes [Dad95, Theorem A] from commutative algebras to residually finite dimensional algebras.

Theorem 4.3 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 unitary 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: The proof is similar to the proof of the previous theorem. One replaces d_{ι} by γ = π_{1}⊕ π_{2}⊕⋯ regarded as a faithful scalar representation of infinite multiplicity into M(K(H)⊗B), and “compress” φ ⊕ γ ≊ ψ ⊕ γ by p_{n} = 1 ⊕ γ_{n}(1). ■

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Marius Dădărlat 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