NUCLEARITY OF REDUCED AMALGAMATED FREE PRODUCT C^{∗}-ALGEBRAS

NARUTAKA OZAWA

Abstract. We show that the reduced free product between two nuclear C^{∗}-algebras is nuclear provided that at least one of the states is pure. We also show that some reduced amalgamated free product C^{∗}-algebras are nuclear. These generalize an unpublished work of Dykema and Smith and verify Dykema’s conjecture. Our proof is modeled after Dykema’s work on exactness of reduced amalgamated free product C^{∗}-algebras.

1. Introduction

The reduced amalgamated free product of C^{∗}-algebras was introduced by Voiculescu [Vo] [VDN]. Dykema [Dy] proved that the reduced amalgamated free product of exact C^{∗}-algebras is exact. We will follow his paper [Dy] the notation. Dykema and Smith [DS] proved, among other things, that if ω is a pure state on the full matrix algebra 𝕄_{n} and ϕ is a state on a nuclear C^{∗}-algebra A, then the reduced free product (𝕄_{n}, ω) ∗ (A, ϕ) is nuclear. We generalize their result and verify Dykema’s conjecture that the reduced free product between two nuclear C^{∗}-algebras is nuclear provided that at least one of the states is pure.

Theorem 1.1. Let B be a unital C^{∗}-algebra and let A_{i} (i = 1, 2) be a unital nuclear C^{∗}-algebra containing B as a unital C^{∗}-algebra and having a conditional expectation ϕ_{i}, from A_{i} onto B, whose GNS representation is faithful. Let (A, ϕ) = (A_{1}, ϕ_{1}) ∗ (A_{2}, ϕ_{2}) be the reduced amalgamated free product of C^{∗}-algebras. Suppose either (i) ϕ_{1} is a pure state (and B = ℂ1) or (ii) 𝕂(E_{1}) ⊂ A_{1} in its GNS representation. Then, A is nuclear.

We remark that the same assertion holds if one replaces nuclearity with one of the following, the LLP, the WEP and exactness. In particular, we recover Dykema’s theorem [Dy].

We recall that a C^{∗}-algebra A is nuclear if A ⊗ C = A ⊗_{max} C for any C^{∗}-algebra C. It is well known that the class of nuclear C^{∗}-algebras is closed under (i) passing to a C^{∗}-subalgebra that is a range of a conditional expectation, (ii) passing to a quotient and (iii) taking an extension.

We recall the definition of reduced amalgamated free product. See [Dy] [Vo] for the detail. Let B be a unital C^{∗}-algebra and let A_{i} (i = 1, 2) be a unital nuclear C^{∗}-algebra containing B as a unital C^{∗}-algebra and having a conditional expectation ϕ_{i} from A_{i} onto B such that for any nonzero a ∈ A_{i}, there is x ∈ A_{i} with ϕ_{i}(x^{∗}a^{∗}ax) ≠ 0. Let ⟨·, ·⟩_{E_{i}} be the B-valued inner product on A_{i}, given by ⟨a, b⟩_{E_{i}} = ϕ_{i}(a^{∗}b), and let E_{i} be the Hilbert B-module obtained from A_{i} by separation and completion. We denote by â the element in E_{i} arising from a in A_{i}. Then A_{i} is faithfully represented on E_{i} by ab̂ = âb. Let ξ_{i} = ˆ1_{A_{i}} ∈ E_{i} be the distinguished element and let E_{i}^{∘} be the complementing B-submodule of ξ_{i}B in E_{i}, i.e., E_{i} = ξ_{i}B ⊕ E_{i}^{∘}. Then the reduced amalgamated free product C^{∗}-algebra (A, ϕ) = (A_{1}, ϕ_{1}) ∗ (A_{2}, ϕ_{2}) is a C^{∗}-subalgebra of 𝔹(E) generated by copies of A_{1} and A_{2}, where

E = ξB ⊕ ⨁_{\begin{matrix}n∈ℕ, i_{1}... ,i_{n}∈\{1,2\}, \\ i_{1}≠i_{2},··· ,i_{n−1}≠i_{n} \end{matrix}} E_{i_{1}}^{∘} ⊗_{B} ⋯ ⊗_{B} E_{i_{n}}^{∘}.

Date: September 8, 2001.

Here ξB is the C^{∗}-algebra B regarded as a Hilbert B-module with the distinguished element ξ = 1_{B}. For a ∈ A_{i}^{∘} = A_{i} ∩ ker ϕ_{i} and for ζ_{j} ∈ E_{i_{j}}^{∘} (j = 1, …, n, n ≥ 2) with i_{1} ≠ ⋯ ≠ i_{n}, we have

a(ζ_{1} ⊗ ⋯ ⊗ ζ_{n}) = \begin{cases}\begin{matrix}(aζ_{1} − ξ_{i}⟨ξ_{i}, aζ_{1}⟩) ⊗ ζ_{2} ⊗ ⋯ ⊗ ζ_{n} \\ +⟨ξ_{i}, aζ_{1}⟩ζ_{2} ⊗ ⋯ ⊗ ζ_{n} \end{matrix} & if i_{1} = i \\ â ⊗ ζ_{1} ⊗ ⋯ ⊗ ζ_{n} & if i_{1} ≠ i. \end{cases}

The reduced amalgamated free product C^{∗}-algebra A is the closed linear span of B and the elements of the form a_{1} ⋯ a_{n} where n ∈ ℕ, a_{j} ∈ A_{i_{j}}^{∘} with i_{1} ≠ ⋯ ≠ i_{n}.

Acknowledgment. The author thank Professor Kenneth Dykema for helpful comments. Most part of this work was carried out while the author was visiting Texas A&M University in summer 2001. He gratefully acknowledges the kind hospitality from Texas A&M University.

2. Proof of Theorem 1.1

The case (i) immediately follows from Choda’s proposition [Ch] (see also [BD]), Kishimoto and Sakai’s lemma [KS] and the result of Dykema and Smith [DS] (or the case (ii) of our theorem). We now concentrate our attention on the case (ii) and assume that 𝕂(E_{1}) ⊂ A_{1}. We may assume that A_{1} and A_{2} are separable thanks to Blanchard and Dykema’s theorem [BD].

Following [Dy], we will construct, in this section, a sequence of ucp maps which approximates the identity map on A. Let p_{1} ∈ 𝔹(E_{1}) be the projection from E_{1} onto ξ_{1}B. Then p_{1} ∈ A_{1} by the assumption. For ζ_{j} ∈ E_{i_{j}}^{∘} (j = 1, …, n, n ≥ 2) with i_{1} ≠ ⋯ ≠ i_{n}, we have

if a ∈ p_{1}A_{1}(1 − p_{1}), then  a(ζ_{1} ⊗ ⋯ ⊗ ζ_{n}) = \begin{cases}⟨ξ_{i}, aζ_{1}⟩ζ_{2} ⊗ ⋯ ⊗ ζ_{n} & if i_{1} = 1 \\ 0 & if i_{1} = 2, \end{cases}

if a ∈ (1 − p_{1})A_{1}(1 − p_{1}), then  a(ζ_{1} ⊗ ⋯ ⊗ ζ_{n}) = \begin{cases}aζ_{1} ⊗ ζ_{2} ⊗ ⋯ ⊗ ζ_{n} & if i_{1} = 1 \\ 0 & if i_{1} = 2, \end{cases}

if a ∈ (1 − p_{1})A_{1}p_{1}, then  a(ζ_{1} ⊗ ⋯ ⊗ ζ_{n}) = \begin{cases}0 & if i_{1} = 1 \\ â ⊗ ζ_{1} ⊗ ⋯ ⊗ ζ_{n} & if i_{1} = 2. \end{cases}

We let, for m, n ≥ 0,

X_{(m,0,n)} = {b_{0}a_{1}b_{1} ⋯ a_{m+n}b_{m+n} : b_{0}, b_{m+n} ∈ ℂ1 ∪ A_{2}^{∘}, b_{1}, …, b_{m+n−1} ∈ A_{2}^{∘},
a_{1} …, a_{m} ∈ (1 − p_{1})A_{1}p_{1}, a_{m+1}, …, a_{m+n} ∈ p_{1}A_{1}(1 − p_{1})}

and

X_{(m,1,n)} = {b_{0}a_{1}b_{1} ⋯ a_{m+n+1}b_{m+n+1} : b_{0}, b_{m+n+1} ∈ ℂ1 ∪ A_{2}^{∘}, b_{1}, …, b_{m+n} ∈ A_{2}^{∘},
a_{1} …, a_{m} ∈ (1 − p_{1})A_{1}p_{1}, a_{m+1} ∈ (1 − p_{1})A_{1}(1 − p_{1}),
a_{m+2}, …, a_{m+n+1} ∈ p_{1}A_{1}(1 − p_{1})}

be the subsets of A.

Lemma 2.1. We have

(1 − p_{1})b(1 − p_{1}) = ϕ_{2}(b)(1 − p_{1})

for b ∈ A_{2}. In particular,

A = \overline{span} ⋃_{m,n≥0}(X_{(m,0,n)} ∪ X_{(m,1,n)}).

Proof. Straightforward. ^{∋}^{∧∧} ⊎ o^{∈} We define Hilbert B-submodules F_{l} of E (l = 0, 1, …) by F_{0} = ξB ⊕ E_{2}^{∘} and

F_{l} = \overbrace{E_{1}^{∘} ⊗_{B} ⋯ ⊗_{B} E_{1}^{∘}}^{2l−1} ⊕ \overbrace{E_{1}^{∘} ⊗_{B} ⋯ ⊗_{B} E_{2}^{∘}}^{2l} ⊕ \overbrace{E_{2}^{∘} ⊗_{B} ⋯ ⊗_{B} E_{1}^{∘}}^{2l} ⊕ \overbrace{E_{2}^{∘} ⊗_{B} ⋯ ⊗_{B} E_{2}^{∘}}^{2l+1}

for l = 1, 2, … and put F_{(→k)} = ⨁_{l=0}^{k} F_{l}. We observe that E = ⨁_{l=0}^{∞} F_{l}. Let Q_{l} (resp. Q_{(→k)}) be the projection from E onto F_{l} (resp. from E onto F_{(→k)}). Lemma 2.2. We have the following.

bQ_{l} = Q_{l}b if b ∈ A_{2},  aQ_{l} = Q_{l−1}a if a ∈ p_{1}A_{1}(1 − p_{1})
aQ_{l} = Q_{l}a if a ∈ (1 − p_{1})A_{1}(1 − p_{1}),  aQ_{l} = Q_{l+1}a if a ∈ (1 − p_{1})A_{1}p_{1}

Proof. Straightforward. ^{∋}^{∧∧} ⊎ o^{∈} We define the isometry V_{k}: E → ℓ_{2} ⊗_{ℂ} F_{(→k)} ⊗_{B} E (k = 0, 1, …) by

V_{k}: \begin{matrix}η \\ ζ_{1} ⊗ ⋯ ⊗ ζ_{m} \\ ζ_{1} ⊗ ⋯ ⊗ ζ_{m} \end{matrix} ⟼ δ_{k−l} ¨⊗η ¨⊗ξ if η ∈ F_{l} with l ≤ k δ_{0} ¨⊗(ζ_{1} ⊗ ⋯ ⊗ ζ_{2k}) ¨⊗(ζ_{2k+1} ⊗ ⋯ ⊗ ζ_{m}) if ζ_{1} ∈ E_{1}^{∘} and m > 2k δ_{0} ¨⊗(ζ_{1} ⊗ ⋯ ⊗ ζ_{2k+1}) ¨⊗(ζ_{2k+2} ⊗ ⋯ ⊗ ζ_{m}) if ζ_{1} ∈ E_{2}^{∘} and m > 2k + 1.

The symbol ¨⊗ is used in order to distingusish various tensor products. Lemma 2.3. The isometries {V_{k}}_{k=0}^{∞} have mutually orthogonal ranges and satisfy the following.

(1 ⊗ b ⊗ 1)V_{k} = V_{k}b if b ∈ A_{2},  (1 ⊗ a ⊗ 1)V_{k} = V_{k−1}a if a ∈ p_{1}A_{1}(1 − p_{1})
(1 ⊗ a ⊗ 1)V_{k} = V_{k}a if a ∈ (1 − p_{1})A_{1}(1 − p_{1}),  (1 ⊗ a ⊗ 1)V_{k} = V_{k+1}a if a ∈ (1 − p_{1})A_{1}p_{1}

Proof. Straightforward. ^{∋}^{∧∧} ⊎ o^{∈} Letting

V_{(→N)} = \frac{1}{\sqrt{N + 1}} ∑_{k=0}^{N} V_{k}

be the isometry from E into ℓ_{2} ⊗_{ℂ} F_{(→N)} ⊗_{B} E, we consider the compression

Φ_{N}: 𝔹(E) → 𝔹(F_{(→N)})

and the ucp map Ψ_{N}: 𝔹(F_{(→N)}) → 𝔹(E) defined by

Ψ_{N}(x) = V_{(→N)}^{∗}(1 ⊗ x ⊗ 1)V_{(→N)}

and we put Θ_{N} = Ψ_{N} ∘ Φ_{N}. Lemma 2.4. The ucp map Θ_{N} maps A into A and

lim_{N→∞} ∥x − Θ_{N}(x)∥ = 0

for every x ∈ A. Indeed, if x ∈ X_{(m,0,n)} ∪ X_{(m,1,n)} then

Θ_{N}(x) = max{0, min{1 − m/(N + 1), 1 − n/(N + 1)}}x.

Proof. This follows from Lemma 2.3. ^{∋}^{∧∧} ⊎ o^{∈} We define the ‘diagonal’ C^{∗}-subalgebra D in A as

D = \overline{span} ⋃_{l≥0}(X_{(l,0,l)} ∪ X_{(l,1,l)}).

This is indeed a C^{∗}-algebra as shown below.

Lemma 2.5. If x ∈ X_{(l,c,l)} and y ∈ X_{(l′,c′,l′)} with l ≥ l′ ≥ 0 and c, c′ ∈ {0, 1}, then

xy ∈ span(X_{(l,c,l)} ∪ X_{(l,1,l)}).

Proof. This follows from Lemma 2.1 and the equation

(1 − p_{1})bab′(1 − p_{1}) = ϕ_{2}(bϕ_{1}(a)b′)(1 − p_{1})

for a ∈ A_{1} and b, b′ ∈ A_{2}^{∘}. ^{∋}^{∧∧} ⊎ o^{∈}

We define the ucp map Δ: A → 𝔹(E) by

Δ(x) = strict- ∑_{l=0}^{∞} Q_{l}xQ_{l}.

Lemma 2.6. The ucp map Δ is a conditional expectation from A onto D and

D = {x ∈ A : xQ_{l} = Q_{l}x for all l = 0, 1, … }.

Proof. This follows from Lemmas 2.1 and 2.2. ^{∋}^{∧∧} ⊎ o^{∈}

Lemma 2.7. The C^{∗}-algebra D is nuclear.

We postpone the proof to Section 3 and now prove Theorem 1.1.

Proof of Theorem 1.1. We may assume that there is a ∈ A_{1}^{∘} (resp. b ∈ A_{2}^{∘}) such that ϕ_{1}(a^{∗}a) = 1 (resp. ϕ_{2}(b^{∗}b) = 1). Indeed, we just replace (B ⊂ A_{1}, ϕ_{1}) with (B ⊗ ℂI ⊂ A_{1} ⊗ 𝕄_{2}, ϕ_{1} ⊗ ψ), where ψ(x) = x_{11}I for x = [x_{ij}] ∈ 𝕄_{2}, and let a = 1 ⊗ e_{21}. We observe that the conditional expectation id ⊗ ψ from A_{1} ⊗ 𝕄_{2} onto A_{1} is compatible, in the sense of [BD], with the conditional expectations ϕ_{1} and ϕ_{1}⊗ψ. The same for A_{2}. Since A = (A_{1}, ϕ_{1})∗(A_{2}, ϕ_{2}) is a C^{∗}-subalgebra of Ã = (A_{1} ⊗𝕄_{2}, ϕ_{1} ⊗ψ)∗(A_{2} ⊗𝕄_{2}, ϕ_{2} ⊗ψ) and is a range of the conditional expectation (ϕ_{1} ⊗ψ)∗(ϕ_{2} ⊗ψ), the nuclearity of A follows from that of Ã. We have used here Theorems 1.3 and 2.2 in [BD].

Let a and b be as above. Then, w = bap_{1} + ab(1 − p_{1}) is an isometry in A with wQ_{l} = Q_{l+1}w for l = 0, 1, … and hence we have wxw^{∗} ∈ D for all x ∈ D. Therefore, A is ∗-isomorphic to the crossed product D by the ∗-endomorphism Ad w and a fortiori is nuclear. ^{∋}^{∧∧} ⊎ o^{∈}

3. Proof of Lemma 2.7

Let D_{(→l)} = Q_{(→l)}D be the C^{∗}-subalgebra of 𝔹(F_{(→l)}). Since the ucp map Θ_{N} appearing in Lemma 2.4 maps D into D and factors through D_{(→N)}, Lemma 2.7 follows from the nuclearity of D_{(→l)} (l = 0, 1, …). Following [Dy], we will prove this by induction.

Lemma 3.1. If x ∈ (⋃_{m≥l+1} X_{(m,0,m)}) ∪ (⋃_{m≥l} X_{(m,1,m)}), then Q_{(→l)}x = 0. In particular,

D_{(→l)} = Q_{(→l)} \overline{span}(( ⋃_{m=0}^{l} X_{(m,0,m)}) ∪ ( ⋃_{m=0}^{l−1} X_{(m,1,m)})).

Proof. Straightforward. ^{∋}^{∧∧} ⊎ o^{∈}

Let

F_{l}^{•} = \overbrace{E_{1}^{∘} ⊗_{B} ⋯ ⊗_{B} E_{2}^{∘}}^{2l} ⊕ \overbrace{E_{2}^{∘} ⊗_{B} ⋯ ⊗_{B} E_{2}^{∘}}^{2l+1}

for l = 0, 1, … and let Q_{l}^{•} be the projection from E onto F_{l}^{•}. We define the unitary W_{l}: F_{l} → F_{l−1}^{•} ⊗_{B} E_{1}^{∘} ⊗_{B} E_{2} (l = 1, 2, …) by

W_{l}: \begin{matrix}ζ_{1} ⊗ ⋯ ⊗ ζ_{2l−1} \\ ζ_{1} ⊗ ⋯ ⊗ ζ_{2l} \\ ζ_{1} ⊗ ⋯ ⊗ ζ_{2l} \\ ζ_{1} ⊗ ⋯ ⊗ ζ_{2l+1} \end{matrix} ⟼ (ζ_{1} ⊗ ⋯ ⊗ ζ_{2l−2}) ¨⊗ζ_{2l−1} ¨⊗ξ_{2} if ζ_{1} ∈ E_{1}^{∘} (ζ_{1} ⊗ ⋯ ⊗ ζ_{2l−2}) ¨⊗ζ_{2l−1} ¨⊗ζ_{2l} if ζ_{1} ∈ E_{1}^{∘} (ζ_{1} ⊗ ⋯ ⊗ ζ_{2l−1}) ¨⊗ζ_{2l} ¨⊗ξ_{2} if ζ_{1} ∈ E_{2}^{∘} (ζ_{1} ⊗ ⋯ ⊗ ζ_{2l−1}) ¨⊗ζ_{2l} ¨⊗ζ_{2l+1} if ζ_{1} ∈ E_{2}^{∘}

and the ucp map σ_{l}: D_{(→(l−1))} → 𝔹(F_{(→(l−1))} ⊕ F_{l}) = 𝔹(F_{(→l)}) by

σ_{l}(x) = x ⊕ W_{l}^{∗}(Q_{l−1}^{•}xQ_{l−1}^{•} ⊗ 1 ⊗ 1)W_{l}

for x ∈ D_{(→(l−1))} ⊂ 𝔹(F_{(→(l−1))}). We define C^{∗}-subalgebras I_{l}^{1} and D_{(→l)}^{•} of D_{(→l)} by

I_{l} = Q_{(→l)} \overline{span} X_{(l−1,1,l−1)}

and

D_{(→l)}^{•} = Q_{(→l)} \overline{span}(( ⋃_{m=0}^{l−1} X_{(m,0,m)}) ∪ ( ⋃_{m=0}^{l−1} X_{(m,1,m)})).

It follows from Lemma 2.5 that these are indeed C^{∗}-algebras and that I_{l} is an ideal of D_{(→l)}^{•}.

Lemma 3.2. The ucp map σ_{l} maps D_{(→(l−1))} into D_{(→l)}^{•}. Indeed, if x ∈ (⋃_{m=0}^{l−1} X_{(m,0,m)}) ∪

(⋃_{m=0}^{l−2} X_{(m,1,m)}), then

σ_{l}(Q_{(→(l−1))}x) = Q_{(→l)}x.

Proof. Straightforward. ^{∋}^{∧∧} ⊎ o^{∈}

Lemma 3.3. If D_{(→(l−1))} is nuclear, then D_{(→l)}^{•} is nuclear.

Proof. Let π_{l} be the quotient map from D_{(→l)}^{•} onto D_{(→l)}^{•}/I_{l} and let ρ_{l} = π_{l} ∘ σ_{l}. We claim that ρ is a ∗-homomorphism from D_{(→(l−1))} onto D_{(→l)}^{•}/I_{l}. In view of Lemma 3.1, it suffices to show

the multiplicativity of ρ_{l} on Q_{(→(l−1))}((⋃_{m=0}^{l−1} X_{(m,0,m)}) ∪ (⋃_{m=0}^{l−2} X_{(m,1,m)})), but this follows from Lemmas 2.5 and 3.2. In particular, D_{(→l)}^{•}/I_{l} is nuclear by the assumption.

We next show that I_{l} is nuclear. Then the nuclearity of D_{(→l)}^{•} follows. As I_{l} = Q_{l}I_{l} by Lemma 3.1, I_{l} is ∗-isomorphic to a C^{∗}-subalgebra of 𝔹(F_{l−1}^{•} ⊗_{B} E_{1}^{∘} ⊗_{B} E_{2}) via the unitary W_{l}. We claim that this C^{∗}-subalgebra is (𝕂(F_{l−1}^{•}) ▷◁ (1 − p_{1})A_{1}(1 − p_{1})) ⊗ ℂ1 and thus I_{l} is nuclear. See [Dy] for the definition and the property of the operation ▷◁. Indeed, if x ∈ X_{(l−1,1,l−1)} is, e.g., of the form

x = b_{0}a_{1}b_{1} ⋯ b_{2l−2}a_{2l−1}b_{2l−1}

with a_{1}, …, a_{l−1} ∈ (1 − p_{1})A_{1}p_{1}, a_{l} ∈ (1 − p_{1})A_{1}(1 − p_{1}), a_{l+1} …, a_{2l−1} ∈ p_{1}A_{1}(1 − p_{1}) and b_{0}, …, b_{2l−1} ∈ A_{2}^{∘}, then

W_{l}xW_{l}^{∗} = (θ_{b̂_{0}⊗â_{1}⊗···⊗b̂_{l−1}}a_{l}θ_{{ \atop b_{2l−1}^{∗}}⊗{ \atop a_{2l−1}^{∗}}⊗···⊗b̂_{l}^{∗}}^{∗}) ⊗ 1.

The other cases are similar and this completes the proof. ^{∋}^{∧∧} ⊎ o^{∈}

We define the ideal J_{l} of D_{(→l)} by

J_{l} = Q_{(→l)} \overline{span} X_{(l,0,l)}.

It follows from Lemmas 2.5 and 3.1 that this is indeed an ideal.

Lemma 3.4. If D_{(→(l−1))} is nuclear, then D_{(→l)} is nuclear.

Proof. Assume that D_{(→(l−1))} is nuclear. Then the nuclearity of D_{(→l)} follows from that of J_{l} thanks to Lemma 3.3 and the fact D_{(→l)} = D_{(→l)}^{•} + J_{l}. As J_{l} = Q_{l}J_{l} by Lemma 3.1, J_{l} is ∗-isomorphic to a C^{∗}-subalgebra of 𝔹(F_{l−1}^{•} ⊗_{B} E_{1}^{∘} ⊗_{B} E_{2}) via the unitary W_{l}. We claim that this C^{∗}-subalgebra is 𝕂(F_{l−1}^{•} ⊗_{B} E_{1}^{∘}) ▷◁ A_{2} and thus J_{l} is nuclear. Indeed, if x ∈ X_{(l,0,l)} is, e.g., of the form

x = b_{0}a_{1}b_{1} ⋯ b_{2l−1}a_{2l}b_{2l}

with a_{1}, …, a_{l} ∈ (1 − p_{1})A_{1}p_{1}, a_{l+1} …, a_{2l} ∈ p_{1}A_{1}(1 − p_{1}) and b_{0}, …, b_{2l} ∈ A_{2}^{∘}, then

W_{l}xW_{l}^{∗} = θ_{b̂_{0}⊗â_{1}⊗···⊗â_{l}}b_{l}θ_{b̂_{2l}^{∗}⊗â_{2l}^{∗}⊗···⊗â_{l+1}^{∗}}^{∗}.

The other cases are similar and this completes the proof. ^{∋}^{∧∧} ⊎ o^{∈}

Proof of Lemma 2.7. As we mentioned, it suffices to show that D_{(→l)} is nuclear for l = 0, 1, …, but this follows from Lemma 3.4 and the fact that D_{(→0)} = A_{2} is nuclear. ^{∋}^{∧∧} ⊎ o^{∈}

Appendix A. An Elementary Proof of the BDS Theorem on Topological Entropy of Free Product Automorphisms

We give an elementary proof of the following theorem of Brown, Dykema and Shlyakhtenko [BDS]. What they actually proved is more general, namely the same assertion for amalgamated free products with finite dimensional amalgams. Consult [Br] for the definition of the topological entropy.

Theorem A.1. Let A_{i} (i = 1, 2) be a unital exact C^{∗}-algebra with a ∗-automorphism α_{i} and let α = α_{1} ∗ α_{2} be the free product ∗-automorphism (defined in [Ch]) on the reduced free product C^{∗}- algebra A = A_{1} ∗ A_{2}. Then, we have

ht(α) = max{ht(α_{1}), ht(α_{2})},

where ht is the Brown-Voiculescu topological entropy.

The Kadison-Schwarz inequality implies that a ucp map ϕ satisfies

∥ϕ(x^{∗}y) − ϕ(x)^{∗}ϕ(y)∥ ≤ ∥ϕ(x^{∗}x) − ϕ(x)^{∗}ϕ(x)∥^{1/2}∥ϕ(y^{∗}y) − ϕ(y)^{∗}ϕ(y)∥^{1/2}

for any x and y. The following Kolmogorov-Sinai type result is a consequence of this inequality.

Lemma A.2. If ω is a finite subset in A with the property that both x^{∗} and x^{∗}x ∈ ω whenever x ∈ ω and A is generated (as a C^{∗}-algebra) by ⋃_{n∈ℤ} α^{n}(ω), then we have ht(α) = ht(α, ω).

Proof of Theorem. We deal with the following situation: A_{i} ⊂ 𝔹(H_{i}) is a C^{∗}-subalgebra and ξ_{i} ∈ H_{i} is a cyclic unit vector. We are given a finite selfadjoint subset ω_{i} of unitaries in A_{i} and ucp maps α_{i}: A_{i} → 𝕄_{r_{i}} and β_{i}: 𝕄_{r_{i}} → 𝔹(H_{i}) such that ∥β_{i} ∘ α_{i}(a) − a∥ < ε for a ∈ ω_{i}. Fix N ∈ ℕ.

Since α_{i} extends to a ucp map from 𝔹(H_{i}) and any ucp map from a von Neumann algebra into a matrix algebra can be approximated by normal ones (in the point-norm topology), we may assume that α_{i} = α′_{i} ∘ Φ_{i}, where Φ_{i}: 𝔹(H_{i}) → 𝔹(H′_{i}) is the compression onto a finite dimensional subspace H′_{i} ⊂ H_{i} and α′_{i}: 𝔹(H′_{i}) → 𝕄_{r_{i}} is a ucp map. Applying the Stinespring theorem to α′_{i} and β_{i}, we obtain an isometry S_{i}: ℓ_{2}^{r_{i}} → H_{i} ⊗ H_{i}^{S} (with some Hilbert space H_{i}^{S}) such that α_{i}(x) = S_{i}^{∗}(x ⊗ 1_{H_{i}^{S}})S_{i} for all x ∈ 𝔹(H_{i}), and an isometry T_{i}: H_{i} → ℓ_{2}^{r_{i}} ⊗ H_{i}^{T} (with some Hilbert space H_{i}^{T}) such that β_{i}(x) = T_{i}^{∗}(x ⊗ 1_{H_{i}^{T}})T_{i}. Defining the isometry V_{i}: H_{i} → H_{i} ⊗ K_{i} with K_{i} = H_{i}^{S} ⊗ H_{i}^{T} by V_{i} = (S_{i} ⊗ 1_{H_{i}^{T}}) ∘ T_{i}, we have ∥V_{i}^{∗}(a ⊗ 1_{K_{i}})V_{i} − a∥ < ε. for a ∈ ω_{i}.

We define ‘length’ of vectors in H_{i} as in the group case: define subspaces by E_{i}^{0} = ℂξ, E_{i}^{k} = span ω_{i}E_{i}^{k−1} for k = 1, …, N − 1 and E_{i}^{N} = H_{i}, and put F_{i}^{k} = E_{i}^{k} ⊖ E_{i}^{k−1} for k = 1, …, N.

Let (H, ξ) = (H_{1}, ξ_{1}) ∗ (H_{2}, ξ_{2}) and K = ℂξ_{K} ⊕ K_{1} ⊕ K_{2} and let χ_{k} = N^{−1/2} ∑_{p=1}^{k} δ_{p} ∈ ℓ_{2}(ℕ). We define an isometry

V : H → H ⊗ K ⊗ H ⊗ ℓ_{2}(ℕ)

by

V ξ = ξ ⊗ ξ_{K} ⊗ ξ ⊗ χ_{N},

and, for ζ_{1} ⊗ ⋯ ⊗ ζ_{l} ∈ F_{i_{1}}^{k_{1}} ⊗ ⋯ F_{i_{l}}^{k_{l}},

V ζ_{1} ⊗ ⋯ ⊗ ζ_{n} =V_{i_{1}}ζ_{1} ⊗ (ζ_{2} ⊗ ⋯ ⊗ ζ_{l}) ⊗ χ_{k_{1}}
+ (ζ_{1} ⊗ V_{i_{2}}ζ_{2}) ⊗ (ζ_{3} ⊗ ⋯ ⊗ ζ_{l}) ⊗ χ_{k_{2}}
+ ⋯ + (ζ_{1} ⊗ ⋯ ⊗ ζ_{d−2} ⊗ V_{i_{d−1}}ζ_{d−1}) ⊗ (ζ_{d} ⊗ ⋯ ⊗ ζ_{l}) ⊗ χ_{k_{d−1}}
+ (ζ_{1} ⊗ ⋯ ⊗ ζ_{d−1} ⊗ V_{i_{d}}ζ_{d}) ⊗ (ζ_{d+1} ⊗ ⋯ ⊗ ζ_{l}) ⊗ χ_{N−(k_{1}+···+k_{d−1})}

if k_{1} + ⋯ + k_{d−1} < N ≤ k_{1} + ⋯ + k_{d}, (here ζ_{d+1} ⊗ ⋯ ⊗ ζ_{l} in the last term should be ξ when d = l) and

V ζ_{1} ⊗ ⋯ ⊗ ζ_{n} =V_{i_{1}}ζ_{1} ⊗ (ζ_{2} ⊗ ⋯ ⊗ ζ_{l}) ⊗ χ_{k_{1}}
+ ⋯ + (ζ_{1} ⊗ ⋯ ⊗ ζ_{l−1} ⊗ V_{i_{l}}ζ_{l}) ⊗ ξ ⊗ χ_{k_{l}}
+ (ζ_{1} ⊗ ⋯ ⊗ ζ_{l}) ⊗ ξ_{K} ⊗ ξ ⊗ χ_{N−(k_{1}+···+k_{l})}

if k_{1} + ⋯ + k_{l} < N. We note that ζ_{1} ⊗ V_{i_{2}}ζ_{2} ∈ H_{i_{1}}^{∘} ⊗ H_{i_{2}} ⊗ K_{i_{2}}, e.g., should be understood as an element in H ⊗ K by the usual convention that H_{i}^{∘} ⊗ ℂξ_{j} ∋ ζ ⊗ ξ_{j} = ζ ∈ H_{i}^{∘} ⊂ H.

Let Φ: 𝔹(H) → 𝔹(H) be the ucp map given by Φ(a) = V^{∗}(a ⊗ 1_{K⊗H⊗ℓ_{2}(ℕ)})V. Then, a brute force computation shows that

∥Φ(a) − a∥ < 3(ε + \frac{1}{N})

for a ∈ ⋃_{i=1}^{m} ω_{i}. Here, we used the crucial fact that aF_{i}^{k} ⊂ F_{i}^{k−1} ⊕ F_{i}^{k} ⊕ F_{i}^{k+1} for a ∈ ω_{i} and k = 1, …, N. Since V_{i}^{∗}(a ⊗ 1_{K_{i}})V_{i} = V_{i}^{∗}(e_{i} ⊗ 1_{H_{i}^{T}})(a ⊗ 1_{H_{i}^{S}} ⊗ 1_{H_{i}^{T}})(e_{i} ⊗ 1_{H_{i}^{T}})V_{i} for the rank r_{i} projection e_{i} of H_{i} ⊗ H_{i}^{S} onto S_{i}ℓ_{2}^{r_{i}}, and dim E_{i}^{N−1} ≤ (|ω| + 1)^{N}, we have the following estimate of the rank r of the ucp map Φ (i.e., Φ factors through 𝕄_{r}):

r ≤ 2N(|ω| + 1)^{N^{2}} max{r_{1}, r_{2}}.

Combined with Lemma A.2, this proves Theorem A.1. ^{∋}^{∧∧} ⊎ o^{∈}

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Department of Mathematical Science, University of Tokyo, Komaba, 153-8914, Japan

   E-mail address: narutaka@ms.u-tokyo.ac.jp