数理解析研究所講究録 1250 巻 2002 年 49-55

49

NUCLEARITY OF REDUCED AMALGAMATED FREE PRODUCTC -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 freeproduct $C$ ’-algebras are nuclear. These generalize an unpublished work of Dykema and Smith andverify Dykema’s conjecture.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 is apure state on the full matrix algebra-algebra , thenand is astate on anuclear
the reduced free productis nuclear. We generalize their result and verify Dykema’s
conjecture that the reduced free product between two nuclear-algebras is nuclear provided thatat least one of the states is pure.
$(i=1, 2)$ be a unital nuclear C’-algebra
Theorem 1.1. Let $B$ be a unital C’-algebra and let
containing $B$ as a unital C’-algebra and having a conditional expectation , fromonto $B$ , whose$GNS$ representation is$(A, \phi)=(\mathrm{A}, \phi_{1})*(A_{2}, \phi_{2})$theamalgamatedLetbereducedfaithful.
freeproduct of C’-algebras. Suppose either (i)) or (ii)
is a pure state (andin
its $GNS$ representation. Then,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 aC’-algebra is nuclear if $A\otimes C=A\otimes_{\max}C$ for any C’-algebra $C$ . It is wellknown that the class of nuclear ’-algebras is closed under (i) passing to aC’-subalgebra that is arange of aconditional expectation, (ii) passing to aquotient and (iii) taking an extension.
We recall the definition of reduced amalgamated free product. See [Dy] [ げ] for the detail. Let$B$ be aunital$(i=1, 2)$ be aunital nuclear C’-algebra containing $B$ as a-algebra and let
unital -algebra and having aconditional expectation fromonto $B$ such that for any nonzero$a\in A_{i}$ , there is $x\in A_{i}$ with $\phi_{i}(x^{*}a^{*}ax)\neq 0$ .
,be the $B$-valued inner product onLet, and letgiven byby separation
be the Hilbert -module obtained from. Thenand completion. We denote by \^a the element inarising from in
is faithfully
represented on
by $ab=ab$ . Let
be the distinguished element and let E7 be the, i.e.,
complementing $B$-submodule ofin. Then the reduced amalgamated freeproduct C’-algebra $(A, \phi)=(A_{1}, \phi_{1})*(A_{2}, \phi_{2})$ is aC’-subalgebra ofgenerated by copies ofand A2, where$A$$\mathrm{C}^{*}$$\phi$$\mathrm{M}_{n}$$\omega$$(\mathrm{M}_{n}, \omega)*(A, \phi)$$C^{*}$$A_{i}$$\phi_{i}$$B=\mathbb{C}1$$\phi_{1}$$A_{i}$$\mathrm{K}(E_{1})\subset A_{1}$

$A$

$A$

$\mathrm{C}$$\mathrm{C}^{*}$$A_{i}$$C^{*}$$\phi_{i}$$\langle\cdot, \cdot\rangle_{E}$$(\mathrm{a}, b\rangle_{E}.\cdot=\phi_{i}(a’ b)$$B$$E_{i}$.

$E_{i}$$A_{i}$$A_{i}$$A_{i}$$A_{i}$$a$$\xi_{i}=\overline{1_{A_{i}}}\in E_{i}$$E_{i}$$\xi_{i}B$$E_{i}$$E_{i}=\xi_{i}B\oplus E_{i}^{\mathrm{o}}$$\mathrm{B}(E)$$A_{1}$$E=\xi B\oplus$$\oplus$$E_{i_{1}}^{\mathrm{o}}\otimes_{E}\cdots\otimes_{B}E_{i_{n}}^{\mathrm{o}}$,$n_{i_{1}\neq i_{2},\cdots,i_{n-1}\neq i_{n}}\in \mathrm{N},i_{1}\ldots,i_{n}\in\{1,2\}$

Date: September 8, 2001.

$A_{i}$

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NARUTAKA OZAWA

Here
For

$\xi B$is the$\mathrm{C}^{*}$-algebra $B$ regarded as aHilbert $B$-module with the distinguished element, we have$(j=1, \ldots, n, n\geq 2)$ with:and for$a\in A:=A:\cap \mathrm{k}\mathrm{e}\mathrm{r}\phi$$\langle$$=1_{B}$.

$i_{1}\neq\cdots\neq \mathrm{i}\mathrm{n}$$\zeta_{j}\in E_{1\mathrm{j}}^{\mathrm{o}}$$(a\zeta_{1}-\xi:\langle\xi:, a\zeta_{1}\rangle)\otimes\zeta_{2}\otimes\cdots\otimes\zeta_{n}$if $i_{1}=i$$a(\zeta_{1}$$(\ \cdots\otimes\zeta_{n})=\{$$+(\xi_{\dot{1}},$ $a\zeta_{1}\rangle\zeta_{2}\otimes\cdots\otimes\zeta_{n}$if$\hat{a}\otimes\zeta_{1}\otimes\cdots\otimes\zeta_{n}$The reduced amalgamated free product,of the form
where$n\in \mathrm{N}$$a_{1}\cdots$ $a_{n}$$\mathrm{C}^{*}$$i_{1}\neq i$.

-algebraA is the closed linear span of $B$ and the elements.
with$i_{1}\neq\cdots\neq i_{n}$$a_{j}\in A_{\dot{\iota}_{\mathrm{j}}}^{\mathrm{o}}$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 ffom Choda’s proposition [Ch] (see also [BD]), Kishimoto andSakai’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
. We may assume thatand A2 are separable thanks to Blanchard and Dykema’s theorem [BD].
Following [Dy], we will construct, in this section, asequence of ucp maps which approximatesthe identity map on . Let
be the projection ffom
. Thenonto
by theassumption. For, we have$(j=1, \ldots, n, n\geq 2)$ with$\mathrm{K}(E_{1})\subset A_{1}$$A_{1}$$A$$p_{1}\in \mathrm{B}(E_{1})$$E_{1}$$p_{1}\in A_{1}$$\xi_{1}B$$i_{1}\neq\cdots\neq i_{n}$$\zeta_{j}\in E_{\dot{|}\mathrm{j}}^{\mathrm{o}}$$\{\xi:$if $a\in p_{1}A_{1}(1-p_{1})$ , then$\mathrm{a}(\mathrm{C}\mathrm{i}\otimes\cdots\otimes(_{f*})$

$=\{$

,

$a\zeta_{1}\rangle\zeta_{2}\otimes\cdots\otimes\zeta_{n}$0if $i_{1}=2$ ,$a\zeta_{1}\otimes\zeta_{2}\otimes\cdots\otimes\zeta_{n}$if $a\in(1-p_{1})A_{1}(1-p_{1})$ , then$\mathrm{a}(\mathrm{C}\mathrm{i}\otimes\cdots\otimes \mathrm{C}_{n})$if $a\in(1-p_{1})A_{1}p_{1}$ , then$=\{$$\mathrm{a}(\mathrm{C}\mathrm{i}\otimes\cdots\otimes\zeta_{n})=\{$$\mathrm{i}\mathrm{f}:_{1}=1$

if $i_{1}=1$

0if $i_{1}=2$ ,0if $i_{1}=1$
if $i_{1}=2$ .

$\hat{a}\otimes\zeta_{1}\otimes\cdots\otimes\zeta_{n}$We let, for m, n$\geq 0$,$X(m,0,n)=\{b_{0}a_{1}b_{1}\cdots$ $a_{m+n}b_{m+n}$:$b_{0}$,$b_{m+n}\in \mathrm{C}1$$a_{1}\ldots$$\cup A_{2}^{\mathrm{O}}$,$b_{1}$,$\ldots$,$b_{m+n-1}\in A_{2}^{\mathrm{O}}$, $a_{m}\in(1-p_{1})A_{1}p_{1}$ , $a_{m+1}$ ,$\ldots$,, $a_{m+n}\in p_{1}A_{1}(1-p_{1})\}$and$X(m,1,n)=\{b_{0}a_{1}b_{1}\cdots a_{m+n+1}b_{m+n+1}$:$b_{0}$, $b_{m+n+1}\in \mathrm{C}1$

be the subsets of$A$,$b_{1}$,$\ldots$,$b_{m+n}\in A_{2}^{\mathrm{O}}$,, $a_{m}\in(1-p_{1})A_{1}p_{1}$ , $a_{m+1}\in(1-p_{1})A_{1}(1-p_{1})$ ,$a_{1}\ldots$$a_{m+2}$$\cup A_{2}^{\mathrm{o}}$,$\ldots$, $a_{m+n+1}\in p_{1}A_{1}(1-p_{1})\}$.

Lemma 2.1. We have$(1-p_{1})b(1-p_{1})=\phi_{2}(b)(1-p_{1})$for$b\in A_{2}$. In particular,$A=\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}\cup(X_{(m,0,n)}\cup X_{(m,1,n)})m,n\geq 0^{\cdot}$

Proof. Straightforward.

$\mathrm{a}$51
NUCLEARITY OF REDUCED AMALGAMATED FREE PRODUCTWe define Hilbert$B$-submodules

of $E$ $(l=0,1, \ldots)$ by$F_{l}$$\underline{2l}$$\underline{2l-1}$

and$F_{0}=\xi B\oplus E_{[mathring]_{2}}$$\underline{2l}$$\underline{2l+1}$$F_{l}=E_{1}^{\mathrm{o}}\otimes_{B}\cdots\otimes_{B}E_{1}^{\mathrm{o}}\oplus E_{1}^{\mathrm{o}}\otimes_{B}\cdots\otimes_{B}E_{2}^{\mathrm{o}}\oplus E_{2}^{\mathrm{o}}\otimes_{B}\cdots\otimes_{B}E_{1}^{\mathrm{o}}\oplus E_{2}^{\mathrm{o}}\otimes_{B}\cdots\otimes_{B}E_{2}^{\mathrm{O}}$for $l=1,2$ ,and put $F(arrow k)=\oplus_{l=0}^{k}F_{l}$ . We observe thatthe projection from $E$ onto
(resp. from $E$ onto $F(arrow k)$ ).
$\ldots$$E=\oplus_{l=0}^{\infty}F_{l}$. Let$Q_{l}$(resp. $Q(arrow k)$ ) be

$F_{l}$

Lemma 2.2. We have the following.
$bQ\iota$$=Q\iota b$$aQ\iota$$=Q\iota a$,
if
if $a\in(1-p_{1})A_{1}(1-p_{1})$ ,$b\in A_{2}$$aQ_{l}$a$=Q_{l}$if $a\in p_{1}A_{1}(1-p_{1})$$aQ_{l}=Q_{l+1}a$$\iota f$$a\in(1-p_{1})A_{1}p_{1}$Proof. Straightforward.
We define the isometryy7$V_{k}$:$\ni^{f}\not\in\phi$$V_{k}$:$Earrow\ell_{2}$Oc$F_{(arrow k)}\otimes_{B}E$$(k=0,1, \ldots)$byif$\delta_{k-l}\otimes\cdot\cdot\eta\otimes\cdot\cdot\xi$$\zeta_{1}\otimes\cdots\otimes\zeta_{m}-\mathrm{J}_{0}\otimes\cdot\cdot(\zeta_{1}\otimes\cdots\otimes\zeta_{2k})\otimes\cdot\cdot(\zeta_{2k+1}\otimes\cdots\otimes\zeta_{m})$$\zeta_{1}\otimes\cdots\otimes\zeta_{m}$The symbol.
$\otimes\cdot$$\delta 0\otimes\cdot\cdot(\zeta_{1}$$(\ \cdots\otimes\zeta_{2k+1})\otimes\cdot\cdot(\zeta_{2k+2}\otimes\cdots\otimes\zeta_{m})$ifif$\eta\in F_{l}$with $l\leq k$$\zeta_{1}\in E_{1}^{\mathrm{o}}$$\zeta_{1}\in E_{2}^{\mathrm{o}}$and $m>2k$
and $m>2k+1$ .

is used in order to distingusish various tensor products.

Lemma 2.3. The isometrries$\{V_{k}\}_{k=0}^{\infty}$

have mutually orthogonal ranges and satisfy the following.

,if$a\in(1-p_{1})A_{1}(1-p_{1})$,if$(1\otimes b\otimes 1)V_{k}=V_{k}b$$b\in A_{2}$$(1\ a\otimes 1)\mathrm{V}\mathrm{k}=V_{k}a$$(1\otimes a\otimes 1)V_{k}$$=V_{k}$a$(1\otimes a\otimes 1)\mathrm{V}\mathrm{k}=V_{k+1}a$

Proof. Straightforward.

if $a\in p_{1}A_{1}(1-p_{1})$
if $a\in(1-p_{1})A_{1}p_{1}$$\neq\in\alpha\backslash$

Letting$V_{(arrow N)}= \frac{1}{\sqrt{N+1}}\sum_{k=0}^{N}V_{k}$be the isometry from $E$ into$\ell_{2}\otimes_{\mathrm{C}}F_{(arrow N)}\otimes_{B}E$$\Phi_{N}$and the ucp map$\Psi_{N}$:$\mathrm{B}(F(arrow N))arrow \mathrm{B}(E)$:, we consider the compression$\mathrm{B}(E)arrow \mathrm{B}(F_{(arrow N\rangle})$defined by$\Psi_{N}(x)=V_{(arrow N)}’(1\otimes x\otimes 1)V(arrow N)$

and we put$\ominus_{N}=\Psi_{N}\circ\Phi_{N}$.

Lemma 2.4. The $ucp$ $map\ominus_{N}$ maps

$A$

into

$A$

and

$\lim_{Narrow\infty}||x-\Theta_{N}(x)||=0$

for every $x\in A$ . Indeed, if $x\in X(m,0,n)\cup X(m,1,n)$ then$\Theta_{N}(x)=\max\{0, \min\{1-m/(N+1), 1-n/(N+1)\}\}x$ .

Proof. This follows from Lemma 2.3.
We define the ‘diagonal’$\mathrm{C}^{*}$-subalgebra $D$ in$\mathrm{a}\epsilon l\phi$

$A$

as

$D= \overline{\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}}\bigcup_{\mathrm{t}\geq 0}(X_{(l,0,l)}\cup X_{(l,1,l)})$This is indeed aC’-algebra as shown below.

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NARUTAKA OZAWA

Lemma 2.5. If$x\in X_{(l,\mathrm{c},l)}$and$y\in X_{(l^{J},\mathrm{c}’,l’)}$$with$ $l\geq l’\geq 0$

and , $d$ $\in\{0, 1\}$ , then$c$$xy\in \mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}(X_{(l,c,l)}\cup X_{(l,1,l)})$.

Proof. This follows from Lemma 2.1 and the equation$(1-p_{1})bab’(1-p_{1})=\phi_{2}(b\phi_{1}(a)b’)(1-p_{1})$for $a\in A_{1}$ and ,$b$$b’\in A_{2}^{\mathrm{O}}$We define the ucp map.

$\#^{\mathrm{W}}$$\Delta:Aarrow \mathrm{B}(E)$

by

A(x)

$= \mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{t}-\sum_{l=0}^{\infty}Q_{l}xQ_{l}$.

Lemma 2.6. The $ucp$ rrgap Ais a conditional expectation from
$D=$

$A$

{ $x\in A:xQ_{l}=Q\iota x$ for all $l=0,1$ ,onto $D$ and$\ldots$}.

Proof. This follows ffom Lemmas 2.1 and 2.2.

$\#^{\emptyset}$Lemma 2.7. The C’-algebra $D$ is nuclear.
We postpone the proof to Section 3and now prove Theorem 1.1.

Proof of Theorem 1.1. We may assume that there is(resp.
) such that $\phi_{1}(a^{*}a)=1$(resp. $ 2 $(b^{*}b)=1$ ). Indeed, we just replace $(B\subset A_{1}, \phi_{1})$ with
,
where
for
, and let $a=1\otimes e_{21}$ . We observe that the conditional
expectation
from
onto
is compatible, in the sense of [BD], with the conditional
expectationsand. The same for . Since $A=(A_{1}, \phi_{1})*(A_{2}$ , $2 is a -subalgebra ofand is arange of the conditional expectation,$a\in A_{1}^{\mathrm{o}}$$b\in A_{2}^{\mathrm{o}}$$(B\otimes \mathrm{C}\mathrm{J}\subset A_{1}\otimes \mathrm{M}_{2}, \phi_{1}\otimes\psi)$$\mathrm{A}(\mathrm{x})=x_{11}I$$x=[x_{i\mathrm{j}}]\in \mathrm{M}_{2}$$\mathrm{i}\mathrm{d}\otimes\psi$$\phi_{1}$$A_{1}\otimes \mathrm{M}_{2}$$A_{1}$$\phi_{1}\otimes\psi$$A_{2}$$C^{*}$$)$$\tilde{A}=$$(A_{1}\otimes \mathrm{M}_{2}, \phi_{1}\otimes\psi)*(A_{2}\otimes \mathrm{M}_{2}, \phi_{2}\otimes\psi)$$(\phi_{1}\otimes\psi)*(\phi_{2}\otimes\psi)$the nuclearity of follows ffom that of . We have used here Theorems 1.3 and 2.2 in [BD].
Let and be as above. Then, $w=bap_{1}+ab(1-\mathrm{p}\mathrm{i})$ is an isometry in with $wQ_{l}=Q_{l+1}w$ for$l=0,1$ ,and hence we have $wxw^{*}\in D$ for all $x\in D$ . Therefore,-isomorphic to the crossed
product $D$ by-endomorphism Ad and afortiori is nuclear.
\yen \eta \$A$$\overline{A}$

$b$

$a$

$A$

$A$

$\ldots$$\mathrm{t}\mathrm{h}\mathrm{e}*$$\mathrm{i}\mathrm{s}*$

$w$

3.

LEMMA 2.7$\mathrm{p}_{\mathrm{R}\mathrm{O}\mathrm{O}\mathrm{F}\mathrm{O}\mathrm{F}}$Let $D(arrow l)=Qtarrow\iota)D$ be the-subalgebra of. Since the ucp map
appearing inLemma 2.4 maps $D$ into $D$ and factors through $D(arrow N)$ , Lemma 2.7 follows from the nuclearity of$D(arrow t)(l=0,1, \ldots)$ . Following [Dy], we will prove this by induction.
$\mathrm{C}^{\iota}$$\mathrm{B}(F(arrow l))$$\Theta_{N}$Le mma 3.1. If x $\in(\cup m\geq l+1X(m,0,m))\cup(\bigcup_{m\geq l(m,1,m)}X)$ , then$Q\mathrm{t}arrow\iota$)

x $=0$ . In particular,$D_{(arrow l)}=Q_{(arrow \mathrm{t})}\overline{\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}}((\cup^{l}X_{(m,0,m)})\cup(\cup l-1X_{(m,1,m)}))$

$m=0$

.

$m=0$

Proof. Straightforward.

$\#\mathrm{w}$Let$\underline{2l}$$\underline{2l+1}$$F_{l}^{\cdot}=E_{1}^{\mathrm{o}}\otimes_{B}\cdots\otimes_{B}E_{2}^{\mathrm{o}}\oplus E_{2}^{\mathrm{o}}\otimes_{B}\cdots\otimes_{B}E_{2}^{\mathrm{O}}$53
NUCLEARITY OF REDUCED AMALGAMATED FREE PRODUCT

for $l=0,1$ ,and let$\ldots$$F_{l-1}^{\cdot}\otimes_{B}E_{1}^{\mathrm{O}}\otimes_{B}$be the projection from $E$ onto

$Q_{\dot{l}}$E2 $l=1,2$ ,$\langle$$\ldots)$

C&$\zeta_{1}$:($-(\zeta_{1}\otimes\cdots\otimes\zeta_{2l-1})\otimes\cdot\cdot\zeta_{2l}\otimes\cdot\cdot\xi_{2}$$\zeta_{1}\otimes\cdots\otimes\zeta_{2l+1}$$\sigma_{l}$:$(\ \cdots\otimes\zeta_{2l-2})\otimes\cdot\cdot\zeta_{2l-1}\otimes\cdot\cdot\xi_{2}$$\zeta_{1}$$(\zeta_{1}\otimes\cdots\otimes\zeta_{2l-2})\otimes\cdot\cdot\zeta_{2l-1}\otimes\cdot\cdot\zeta_{2l}$$\cdots\otimes\zeta_{2l}$$\zeta_{1}\otimes\cdots\otimes\zeta_{2l}$and the ucp map. We define the unitary$W_{l}$:$F\iota$$arrow$by$\zeta_{1}\otimes\cdots\otimes\zeta_{2l-1}$$W_{l}$$F_{l}^{\cdot}$$(\zeta_{1}\otimes\cdots\otimes\zeta_{2l-1})\otimes\cdot\cdot\zeta_{2l}\otimes\cdot\cdot\zeta_{2l+1}$

if$\zeta_{1}\in E_{1}^{\mathrm{o}}$if$\zeta_{1}\in E_{1}^{\mathrm{o}}$if$\zeta_{1}\in E_{2}^{\mathrm{o}}$if$\zeta_{1}\in E_{2}^{\mathrm{o}}$by$D_{(arrow(l-1))}arrow \mathrm{B}(F_{(arrow(l-1))}\oplus F_{l})=\mathrm{B}(F_{(arrow l)})$$\sigma\iota(x)=x\oplus W_{l}’(Q_{\dot{l-}1}xQ_{\dot{l-}1}\otimes 1\otimes 1)W_{l}$for$x\in D\mathrm{t}arrow(l-1))\subset \mathrm{B}(F\mathrm{t}arrow(l-1)))$. We define C’-subalgebras$I_{l}=Q(arrow l)$

span

and

$I_{l}^{1}$of$D_{(arrow l)}^{\cdot}$$D\mathrm{t}arrow\iota$)by$X(\downarrow-1,1,l-1)$

and$D_{(arrow l)}^{\cdot}=Q_{(arrow l)}\overline{\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}}((\cup X_{(m,0,m)})\cup(\cup X_{(m,1,m)}))m=0m=0l-1l-1$

It follows ffom Lemma 2.5 that these are indeed C’-algebras and that

Lemma 3.2. The $ucp$ map, then$\sigma\iota$

maps $D(arrow(l-1))$ into.

$D_{(arrow l)}^{\cdot}$

Indeed,$I\iota$.

is an ideal of$D_{(arrow \mathrm{t})}^{\cdot}$.

if $x\in(\cup^{l-1}m=0X(m,0,m))\cup$$( \bigcup_{m=0}^{l-2}X_{(m,1,m)})$$\sigma\iota(Q_{(arrow(l-1))}x)=Q_{(arrow l)^{X}}$Proof. Straightforward.
Lemma 3.3. If $D(arrow(l-1))$ is nuclear, then.
$\#’\phi$$D_{(arrow \mathrm{t})}^{\cdot}$

is nuclear.

. We claim thatand letbe the quotient map fromonto
. In view of Lemma 3.1, it suffices to show
is a -homomorphism fromonto
$Q(arrow(l-1))((\cup^{l-1}m=0X(m,0,m))\cup(\cup^{l-2}m=0X(m,1,m)))$, but this follows fromthe multiplicativity of on
Lemmas 2.5 and 3.2. In particular,
is nuclear by the assumption.
by Lemma
follows. As
We next show that is nuclear. Then the nuclearity ofE2) via the unitary Wj. We claim
3.1,-isomorphic to a
-subalgebra of B(
and thus is nuclear. See [Dy] for-subalgebra isthat thisIndeed,.
$x\in
X(l-1,1,l-1)$ is, e.g., of the form
if
operation
property
and
the
of
the
the definition

Proof. Let$\pi\iota$$\rho$$D\mathrm{t}arrow(l-1))$$\beta \mathrm{t}=\pi_{l}\circ\sigma\iota$$D_{(arrow l)}^{\cdot}/I\iota$$D_{(arrow l)}^{\cdot}$$*$$D_{(arrow \mathrm{t})}^{\cdot}/I_{l}$$\rho\iota$$D_{(arrow l)}^{\cdot}/I_{l}$$I_{l}$$I\iota$$I_{l}=Q\iota I_{l}$$D_{(arrow \mathrm{t})}^{\cdot}$$\mathrm{C}^{*}$$\mathrm{i}\mathrm{s}*$$F_{l-1}^{\cdot}\otimes_{B}E_{1}^{\mathrm{o}}\otimes_{B}$$(\mathrm{K}(F_{l-1}^{\cdot})\mathrm{N} (1-p_{1})A_{1}(1-p_{1}))\otimes \mathrm{C}1$$\mathrm{C}^{*}$$I_{l}$$\triangleright\triangleleft$$x=b_{0}a_{1}b_{1}\cdots$ $b_{2l-2}a_{2l-1}b_{2l-1}$

,
with
,
,
$a_{1}$$b_{0}$$\ldots$$\ldots$, $a\iota-1\in(1-p_{1})A_{1}p_{1}$ ,, then$a_{l}\in(1-\mathrm{p}\mathrm{i})\mathrm{A}\mathrm{i}(1-p_{1})$,$a_{l+1}\cdots$, $a_{2l-1}\in p_{1}A_{1}(1-p_{1})$ and$b_{2l-1}\in A_{2}^{\mathrm{o}}$$W_{lX}W_{\mathrm{t}}’=( \theta_{\hat{b_{\mathrm{O}}}\otimes\overline{a_{1}}\emptyset\cdots\otimes b_{1-1}}-a\iota\theta_{\otimes\overline{a_{\dot{2}l-1}}\otimes\cdots\emptyset\hat{b}}.\frac{*}{b_{21-1}}\iota)\otimes 1$

The other cases are similar and this completes the proof.
We define the ideal$J_{l}$$\#/\wp$

of $D(arrow l)$ by$J_{l}=Q_{(arrow l)}$span

$X_{(l,0,l)}$.

It follows from Lemmas 2.5 and 3.1 that this is indeed an ideal.
Lemma 3.4. If $D(arrow(l-1))$ is nuclear, then$D_{(arrow l)}$is nuclear.

.

54
NARUTAKA OZAWA

thanksis nuclear. Then the nuclearity of) follows from that of
Proof. Assume that
3.1,.
-isomorphic
by
Lemma
to
As
to Lemma 3.3 and the fact)E2) via the unitary Wj. We claim that this C’-subalgebra isaC’-subalgebra of R(
and thus is nuclear. Indeed, if) is, e.g., of the form
$D_{(arrow(l-1))}$$J_{l}$$D\mathrm{t}arrow\iota$$Dtarrow l$$J_{l}=Q_{l}J_{l}$$=D_{(arrow l)}^{\cdot}+J_{l}$$J_{l}\mathrm{i}\mathrm{s}*$$F_{l-1}^{\cdot}\otimes_{B}E_{1}^{\mathrm{O}}\otimes_{B}$$J_{l}$$\mathrm{K}(F_{\dot{l-}1}\otimes_{B}E_{1}^{\mathrm{o}})\triangleright\triangleleft A_{2}$$x\in X\mathrm{t}\iota.0,\iota$$x=b_{0}a_{1}b_{1}\cdots$ $b_{2l-1}a_{2l}b_{2l}$with$a_{1}$,$\ldots$,$a_{\mathrm{t}}\in(1-p_{1})A_{1}p_{1}$,$a_{l+1}\ldots$, $a_{2l}\in p_{1}A_{1}(1-p_{1})$ and$b_{0}$,$\ldots$,$b_{2l}\in A_{2}^{\mathrm{o}}$, then.

$W \iota xW^{*}\iota=\theta f_{0\Phi\overline{a_{1}}\Phi\cdots\Phi\hat{u}\iota\cdot\Phi\overline{a_{\dot{\mathrm{z}}\iota}}\Phi\cdots\Phi a_{\dot{\mathrm{t}+}1}}b_{l}\theta\frac{*}{b_{21}}-$The other cases are similar and this completes the proof.

$\#\mathrm{w}$Pr oof of Le mma 2.
nuclear for l $=0,$ 1,
we mentioned, it suffices to show that$=A_{2}$ is nuclear.
this follows from Lemma 3.4 and the fact that)
$D(arrow l).\mathrm{i}\mathrm{s}$$7\mathrm{A}\mathrm{s}\mathrm{m}$$\ldots,*^{\eta\backslash }\mathrm{b}\mathrm{u}\mathrm{t}$$D\mathrm{t}arrow 0$

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 productswith 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
and letbe the free product -automorphism(defined in [Ch]) on the reduced free product C’algebra $A=A_{1}*A_{2}$ . Then, we have$\alpha.\backslash$$*$$\alpha=\alpha_{1}*\alpha_{2}$,$\mathrm{h}\mathrm{t}(\mathrm{a})=\max\{\mathrm{h}\mathrm{t}(\alpha_{1}), \mathrm{h}\mathrm{t}(\alpha_{2})\}$

where ht is the Brvyum-Voiculescu topological entropy.

The Kadison-Schwarz inequality implies that a ucp map$\phi$

satisfies$||\phi(x^{*}y)-\phi(x)^{*}\phi(y)||\leq||\phi(x^{*}x)-\phi(x)^{*}\phi(x)||^{1/2}||\phi(y^{*}y)-\phi(y)^{*}\phi(y)||^{1/2}$

for any

and . The following Kolmogorov-Sinai type result is aconsequence of this inequality.

$x$

$y$

with the property that both
Lemma A.2. If is a finite subset in, then we haveand is generated (as a C’-algebra) by$A$$\omega$$A$$x\in‘\theta$$x^{*}$$\bigcup_{n\in \mathrm{Z}}\alpha^{n}(\omega)$

Proof of Theorem. We deal with the following situation:

and $x^{*}x\in\omega$ whenever$\mathrm{h}\mathrm{t}(\alpha)=\mathrm{h}\mathrm{t}(\alpha,\omega)$.

is aC’-subalgebra andselfadjoint
afinitesubsetofand ucp maps
is acyclic unit vector. We are given
unitaries in
:
.
.
:
Fix
such thatforandand any ucp map from avon Neumann algebra
Since :extends to aucp map from
into amatrix algebra can be approximated by normal ones (in the point-norm toPology), we may, where:is the compression onto afinite dimensional
assume that:subspace
is aucp map. Applying the Stinespring theorem toand.
(with some Hilbert space) such thatci; and , we obtain an isometry :.
, and an isometry :(with some Hilbert
for all@. Defining the isometry :space) such thatwith$-a||<\epsilon$, we have
. for
.
by V4
as in the group case: define subspaces by
We define length’ of vectors in, $N$ .
span
, $N-1$ and $E_{i}^{N}=H:$ , and put
for $k=1$ ,for $k=1$ ,and let.
Let $(7\{, \xi)=(H_{1}, \xi_{1})*(H_{2}, \xi_{2})$ andWe define an isometry$\alpha_{i}$$A_{:}arrow \mathrm{M}_{r_{j}}$$\beta_{:}$$\mathrm{M}_{r}$.

$A_{:}$$\omega_{\dot{l}}$$||\beta_{\dot{\iota}}\circ\alpha:(a)-a||<\epsilon$$arrow \mathrm{B}(?t:)$$N\in \mathrm{N}$$a\in\omega:$$\mathrm{B}(H:)$$\alpha$$\alpha:=\alpha_{\dot{\iota}}’\circ\Phi:$$H’.\cdot\subset H_{:}$$\alpha_{\dot{1}}’$$\Phi_{:}$$\mathrm{B}(7\{:)arrow \mathrm{B}(H’|.)$$\mathrm{B}(H_{\dot{l}}’)arrow \mathrm{M}_{r}$$s.\cdot$$\beta_{\dot{1}}$$\mathrm{o}\mathrm{a}(\mathrm{x})=S_{\dot{1}}^{*}$$\xi_{:}\in H_{:}$$A_{:}\subset \mathrm{B}(7\{_{\dot{1}})$$H_{\dot{1}}^{T}$$\mathcal{K}_{:}=H_{\dot{\iota}}^{S}\otimes H_{\dot{l}}^{T}$$\Gamma_{2}\cdot$$arrow\gamma\{_{\dot{1}}$$\otimes H_{\dot{1}}^{S}$$x\in \mathrm{B}(\mu_{:})$$(x\otimes 1_{H}\dot{.}s)S_{\dot{l}}$$H_{\dot{1}}^{S}$$T_{\dot{1}}$$\pi_{:}arrow\Gamma_{2}\cdot$$V_{\dot{1}}$$\beta_{\dot{1}}(x)=T_{\dot{1}}^{*}(x\otimes 1_{H^{T}}.)T_{\dot{1}}$$=$$H_{\dot{1}}^{T}$$(S_{\dot{1}} \otimes 1_{H_{}^{T}})\circ T_{\dot{1}}$$E_{\dot{l}}^{0}=\mathrm{C}(,$$\mu_{:}$$\omega_{i}E_{i}^{k-1}$$?\mathrm{t}:arrow H_{:}\otimes \mathcal{K}_{:}$$a\in\omega:$$||V_{\dot{1}}^{*}(a\otimes 1\kappa_{j})V_{\dot{1}}$$F_{\dot{l}}^{k}=E_{\dot{1}}^{k}$$\ldots$$\mathcal{K}=\mathbb{C}\xi\kappa\oplus \mathcal{K}_{1}\oplus \mathcal{K}_{2}$$V:H$$arrow \mathcal{H}\otimes \mathcal{K}\otimes H$$\otimes\ell_{2}(\mathrm{N})$$E_{j}^{k}=$$\ominus E_{\dot{1}}^{k-1}$$\ldots$$\chi_{k}=N^{-1/2}\sum_{p=1}^{k}\delta_{p}\in\ell_{2}(\mathrm{N})$55
NUCLEARITY OF REDUCED AMALGAMATED FREE PRODUCTby$V\xi=\xi\otimes\xi_{\mathcal{K}}\otimes\xi\otimes\chi_{N}$and, for,,,$\zeta_{1}\otimes\cdots\otimes\zeta\iota\in F_{i_{1}}^{k_{1}}\otimes\cdots F_{i}^{k_{1}}$$V\zeta_{1}\otimes\cdots\otimes\zeta_{n}=V_{i_{1}}\zeta_{1}\otimes(\zeta_{2}\otimes\cdots\otimes\zeta_{l})\otimes\chi_{k_{1}}$$+(\zeta_{1}\otimes V_{i_{2}}\zeta_{2})\otimes(\zeta_{3}\otimes\cdots\otimes\zeta_{l})\otimes\chi_{k_{2}}$$+\cdots+(\zeta_{1}\otimes\cdots\otimes\zeta_{d-2}\otimes V_{i_{d-1}}\zeta_{d-1})\otimes(\zeta_{d}\otimes\cdots\otimes\zeta_{l})\otimes\chi_{k_{d-1}}$$+(\zeta_{1}\otimes\cdots\otimes\zeta_{d-1}\otimes V_{i_{d}}\zeta_{d})\otimes(\zeta_{d+1}\otimes\cdots\otimes\zeta_{l})\otimes\chi N-(k_{1}+\cdots+k_{d-1})$

if $k_{1}+\cdots+k_{d-1}<N\leq k_{1}+\cdots+k_{d}$ , (here$\zeta_{d+1}\otimes\cdots\otimes\zeta\iota$

and$V\zeta_{1}\otimes$

in the last term should be 4when $d=l$ )$\cdots\otimes\zeta_{n}=V_{i_{1}}\zeta_{1}\otimes(\zeta_{2}\otimes\cdots\otimes\zeta_{l})\otimes\chi_{k_{1}}$$+\cdots+(\zeta_{1}\otimes\cdots\otimes\zeta_{l-1}\otimes V_{i_{\mathrm{t}}}(_{l})\otimes\xi\otimes\chi_{k_{l}}$$+(\zeta_{1}\otimes\cdots\otimes\zeta_{l})\otimes\xi\kappa\otimes\xi\otimes\chi N-(k_{1}+\cdots+k_{1})$

if $k_{1}+\cdots+k_{l}<N$ . We note thatby the usual convention thatelement inbe the ucp map given by $(a)=V’
Let
computation shows that

, e.g., should be understood as an.
. Then, abrute force$\zeta_{1}\otimes V_{i_{2}}\zeta_{2}\in H_{[mathring]_{1}_{i}}\otimes H_{i_{2}}\otimes \mathcal{K}_{i_{2}}$$\gamma\{_{[mathring]_{i}}\otimes \mathbb{C}\xi j\ni\zeta\otimes\xi_{j}=\zeta\in H_{[mathring]_{i}}\subset H$$7\{\otimes \mathcal{K}$$\Phi:\mathrm{M}(7\mathrm{i})arrow \mathrm{B}(H)$$(a\otimes 1\mathcal{K}\otimes H\otimes\ell_{2}(\mathrm{N}))V$$|| \Phi(a)-a||<3(\epsilon+\frac{1}{nr})$for$a \in\bigcup_{i=1}^{m}\omega_{t}$. Here, we used the crucial fact that$aF_{i}^{k}\subset F_{i}^{k-1}\oplus F_{i}^{k}\oplus F_{i}^{k+1}$

for$a\in\omega_{i}$andfor the rank
, $N$ . Since$\dim
E_{i}^{N-1}\leq(|\omega|+1)^{N}$, we have the following estimate of
, and
onto
of
projection):
factors throughthe rank of the ucp map (i.e.,$k=1$ ,

$V_{i}’(a\otimes 1\kappa_{:})V_{i}=V_{i}^{*}(e_{i}\otimes 1_{H^{T}}.)(a\otimes 1_{H}.\cdot s\otimes 1_{H^{T}}\dot{.})(e_{i}\otimes 1_{H_{i}^{T}})V_{i}$$\ldots$$e_{i}$$H_{i}\otimes H_{i}^{S}$$\Phi$$\Phi$$r$$r_{i}$$S_{i}\ell_{2}^{r}\cdot.$$\mathrm{M}_{r}$$r \leq 2N(|\omega|+1)^{N^{2}}\max\{r_{1}, r_{2}\}$.

Combined with Lemma A2, this proves Theorem A1.

善

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[VDN] D Voiculescu, K.J Dykema, A. Nica, Free random variables, CRM Monograph Series, 1, American Mathematical Society, 1992
DEpARTMENT OF MATHEMATICAL SCIENCE. UNIVERSITY OF Tokyo, 153-8914, JAPAN-mail address: narutaka$ms. -tokyo.

$E$$\mathrm{u}$$\mathrm{a}\mathrm{c}$. jP