数理解析研究所講究録 1291 巻 2002 年 45-54

45

ASURVEY ON CHARACTERIZATIONOF NUCLEAR C -ALGEBRAS*NARUTAKA OZAWA(小沢登高)

1. NUCLEARITY AND INJECTIVITY
We assume the reader to have basic knowledge on tensor products of C’-algebra.
For this, the appendix of [We] is readable. See also [Tal].
A -algebraA is said to be nuclear if one has $A\otimes_{\max}B=A\otimes_{\min}B$ for any . Here,by $A\otimes_{\max}B=A\otimes_{\min}B$ , we mean that the canonical quotient map from-isomorphic. (I am not sure whether it can happen that
onto
is a
are -isomorphic without being injective.) If
andextends:(completely positive) contraction, then the map:
and:to a cp contractions$\mathrm{T}$$B$$C^{*}$$A\otimes_{\max}B$$Q$$A\otimes_{\max}B$$A\otimes_{\min}B\mathrm{i}\mathrm{s}*$$A\otimes_{\min}B$$\varphi:A_{1}arrow A_{2}$

$Q$

$*$

$\varphi\otimes \mathrm{i}\mathrm{d}_{B}$$A_{1} \otimes\min Barrow A_{2}\otimes\min B$$\varphi\otimes\min \mathrm{i}\mathrm{d}_{B}$$\mathrm{c}\mathrm{p}$$A_{1}\otimes_{\mathrm{a}\mathrm{g}}Barrow A_{2}\otimes_{\mathrm{a}}B$$\varphi\otimes_{\max}\mathrm{i}\mathrm{d}_{B}$$A_{1}\otimes_{\max}Barrow$. This fact follows from the Stinespring representation theorem. We usually-algebraA is avonof a
. The second dual
omit the subscript of$A_{2}\otimes_{\max}B$$C^{*}$$A^{**}$$\varphi\otimes \mathrm{i}\mathrm{d}_{B}$is said to be injective if
Neumann algebra. Avon Neumann algebra$M$
onto . This property does not depends on
there is a cp projection from
achoice of faithful normal representations of $M$ . A cp projection is often called aconditional expectation because of the following fact. Let be a -subalgebra ofis multiplicative, then
. If
into
and be acp contraction fromfor , $b\in A$ andis automatically an -bimodule map, i.e.,$x\in B$ . This follows from the Stinespring representation theorem; $\varphi(xy)-\varphi(x)\varphi(y)=$$[V^{*}\pi(x)(1-VV^{*})^{1/2}][(1-VV^{*})^{1/2}\pi(y)V]=X\mathrm{Y}$ and $XX^{*}=\varphi(xx^{*})-\varphi(x)\varphi(x)^{*}$ ,-algebraA is said to have the CPAP (completely
etc. See [Ch] for the detail. A
positive approximation property) if there is anet of finite rank cp contractions on
pointwisely, i.e., limi $||a-\theta_{i}(a)||=0$ for all $a\in|A$ . We often
which converges torequire that factors through full matrix algebra, i.e., there are $n=n(i)\in \mathrm{N}$ and cp. These two CPAP’s are
such thatand :contractions :equivalent as we will see in Theorem 1. The corresponding notion for von Neumann
algebras is semidiscreteness. Avon Neumann algebra $M$ is said to be semidiscrete
if there is anet of normal finite rank ucp (unital completely positive) maps onfor,in the point-a-weak topology,
which converges to
all $a\in M$ . We often require that factors through afull matrix algebra. Since theseproperties which we will deal with are all stable under unitization, we will assume
that all C’-algebras are unital from now on$M\subset \mathrm{B}(\mathcal{H})$$\mathrm{B}(\mathcal{H})$$\varphi$

$A$

$B$

$B$

$\mathrm{B}(\mathcal{H})$$C^{*}$$\varphi|_{A}$$\varphi$$\varphi$$\varphi(axb)=\varphi(a)\varphi(x)\varphi(b)$

$A$

$a$

$C^{*}$$\theta_{:}$$A$$\mathrm{i}\mathrm{d}_{A}$$\theta_{i}$$\sigma_{\dot{l}}$$Aarrow \mathrm{M}_{n}$$\rho_{i}$$\mathrm{M}_{n}arrow A$$\rho.\cdot\sigma_{i}=\theta:$$\theta_{:}$$\mathrm{i}.\mathrm{e}.$$\mathrm{i}\mathrm{d}_{M}$$\theta_{\dot{*}}$$\sigma- \mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}-\lim_{\dot{1}}\theta:(a)=a$

$A$

46
NARUTAKA OZAWA$(’\rfloor\backslash \Re \mathrm{g}\overline{\mathrm{f}\mathrm{f}})$

The following theorem is fundamental in the study of nuclear C’-algebras. The
is due to [EL] and the converseis due to [CE1]. The partis due to [CE2] and [Kil]. We will only prove. Direct
proofs ofare found in [CE2] and [Kil], the latter of which is almost same asin Theorem 2. The implication
the proof ofis due to [C02] and[BP]. The converse implicationis adeep result of Haagerup [Hal] whichuses Connes’ celebrated theorem [Col] (Theorem 2below). We will prove later apoor man’s version of this implication.
part$(\mathrm{i})\Rightarrow(\mathrm{i}\mathrm{i})$$(\mathrm{i}\mathrm{i})\Rightarrow(\mathrm{i})$$(\mathrm{i})\Rightarrow(\mathrm{i}\mathrm{i})\Rightarrow(\mathrm{i}\mathrm{i}\mathrm{i})\Rightarrow(\mathrm{i})$$(\mathrm{i})\Leftrightarrow(\mathrm{i}\mathrm{i}\mathrm{i})$$(\mathrm{i})\Rightarrow(\mathrm{i}\mathrm{i}\mathrm{i})$$(\mathrm{i}\mathrm{v})\Rightarrow(\mathrm{i}\mathrm{i})$$(\mathrm{i}\mathrm{i})\Rightarrow(\mathrm{i}\mathrm{i}\mathrm{i})$$(\mathrm{i})\Rightarrow(\mathrm{i}\mathrm{v})$

Theorem 1. For a C’-algebra A, the following are equivalent(i) The C’-algebra is nuclear.
(ii) The second dual $A’*iS$ injective.
(iii) The C’-algebra has the CPAP.
(iv). The Banach algebra is amenable.
$A$

$A$

$A$

. We follow [La] for the proof. Let
proofsrepresentation. Since is nuclear, the representation$A^{**}\subset \mathrm{B}(\mathcal{H})$$(\mathrm{i})\Rightarrow(\mathrm{i}\mathrm{i})$be afaithful normal$A$$\pi:A\otimes_{\min}A’\ni\sum_{k}a_{k}\otimes x_{k}-\sum_{k}a_{k}x_{k}\in \mathrm{B}(\mathcal{H})$be a ucp extension of and let :B(??). Since is an-bimodule map, is aconditional
expectation fromonto $A’$ . This shows $A’$ , and afortiori $A’=A^{**}$ , is injective.
. It follows from Theorem 2below that injectivity is equivalent to semidiscreteness. We will prove that has the CPAP provided thatis semidiscrete.
By semidiscreteness, the identity map
on
is approximated, in the pointweak’ topology, by finite rank ucp maps which factor through full matrix algebras.
Since aucp map from afull matrix algebra
into
is approximated, in the
point-weak’ topology, by a ucp map from
into (observe that amap from
into aC’-algebra is cp if and only if
is positive), the identity
map
on is approximated, in the point-weak topology, by finite rank ucp maps
which factor through full matrix algebras. Since the point-weak closure of aconvex
set of bounded linear maps on aBanach space coincides with the point-norm closure,
this completes the proof.
. Let:be anet of finite rank ucp maps on which converges topointwisely. Let be aC’-algebra, $Q:A\otimes_{\max}Barrow A\otimes_{\min}B$ be the canonical quotient. We have thatand takesince both maps
. This implies thatare continuous and coincide on.
is of finite rank, we have
Since
. It follows that
. Since the ucp maps
converges to, we have$x=0$ . This shows $A\otimes_{\max}B=A\otimes_{\min}$ B.

is continuous. Let$\Phi:A\otimes_{\min}\mathrm{B}(\mathcal{H})arrow \mathrm{B}(??)$$x\mapsto*\Phi(1\otimes x)\in \mathrm{B}(\mathcal{H})$$\Phi$$\pi$$(A\otimes_{\min}A’)$$\varphi$$\ni$$\varphi$$\mathrm{B}(\mathcal{H})$$(\mathrm{i}\mathrm{i})\Rightarrow(\mathrm{i}\mathrm{i}\mathrm{i})$$A$$A^{**}$$A^{**}$$\mathrm{i}\mathrm{d}_{A’},$$A^{**}$$\mathrm{M}_{n}$$\mathrm{M}_{n}$

$B$

$A$

$\varphi$$\mathrm{M}_{n}$$[\varphi(e_{\dot{l}j})]:,j\in \mathrm{M}_{n}(B)$$A$$\mathrm{i}\mathrm{d}_{A}$$(\mathrm{i}\mathrm{i}\mathrm{i})\Rightarrow(\mathrm{i})$$A$$\{\varphi_{i}\}$$\mathrm{i}\mathrm{d}_{A}$$B$$x\in \mathrm{k}\mathrm{e}\mathrm{r}Q$$(\varphi:\otimes_{\min}\mathrm{i}\mathrm{d}_{B})Q=Q(\varphi:\otimes_{\max}\mathrm{i}\mathrm{d}_{B})$$Q(\varphi:\otimes_{\max}\mathrm{i}\mathrm{d}_{B})(x)=0$$A\otimes_{\mathrm{a}}B$$\varphi_{\dot{*}}$$(\varphi_{\dot{*}}\otimes_{\max}\mathrm{i}\mathrm{d}_{B})(x)=0$$(\varphi:\otimes_{\max}\mathrm{i}\mathrm{d}_{B})(x)\in A\otimes_{\mathrm{a}}\mathrm{g}B$$\varphi_{*}.\otimes_{\max}\mathrm{i}\mathrm{d}_{B}$$\mathrm{i}\mathrm{d}_{A\otimes_{\mathrm{m}\mathrm{r}}B}$$\square$

47
ASURVEYis
The following is acelebrated theorem of Connes [Col]. The part[Ha2](iii).
(ii)
See
and
(i),
among
equivalence
prove
the
due to [EL]. We will only.
and [Po] for simple proofs of$(\mathrm{i}\mathrm{i}\mathrm{i})\Leftrightarrow(\mathrm{i}\mathrm{i})\Rightarrow(\mathrm{i})$$(\mathrm{i})\Rightarrow(\mathrm{i}\mathrm{v})$

Theorem 2. For a von Neumann algebra M(i) The von Neumann algebra $M$ is injective.
(ii) The representation(iii) The von $N$ eumann algebra $M$ is semidiscrete.
(iv). The von Neumann algebra $M$ is $AFD$ .

$\subset \mathrm{B}(?t)$, the following are equivalent$\pi:M\otimes_{\min}M’\ni\sum a_{k}\otimes b_{k}\vdash+\sum a_{k}b_{k}\in \mathrm{B}(\mathcal{H})$Proof.

$(\mathrm{i})\Rightarrow(\mathrm{i}\mathrm{i})$is continuous.

. We follow [Wal] for the proof. We first assume that $M$ is finite withanormal faithful tracial state and let be aconditional expectation fromis astandard
onto $M$ . To prove continuity of , we may assume that
representation. Then, continuity of follows by applying Theorem 3below to thefor $M$ . Now, it is not hard to show continuity of for asemifinite
hypertrace
injective von Neumann algebra. The general case then follows from the Takesaki
duality theorem [Ta2]. See [Wal] for the detail.
be the convex set of all (not. We follow [Kil] for the proof. Let
where is a ucp map fromnecessarily contractive) cp maps 0on $M$ of the form
and is a(not necessarily contractive) cp map from
into afull matrix algebra
is ininto $M$ . It suffices to show that the identity map
the full matrix algebra
coincides
setofaconvexclosure-weak
the point-cr-weak closure of since thewith the poinW-strong closure (hence we can perturb them to unital ones). To prove, $a_{n}\in M$ and $>0$ . We,
on $M$ , ,it, we give ourselves normal states ,, . Let $f=n^{-1} \sum_{k}f_{k}$for all $k=1$ ,withhave to find
, $x_{n}\in\pi f(M)’$be the GNS triplet. It follows that there are ,andsuch that $f_{k}(a)=(\pi_{f}(a)x_{k}\xi_{f}, \xi_{f})$ for $a\in M$ . Let be astate on $M\otimes_{\min}\pi f(M)’$given by $\omega(a\otimes x)=(\pi_{f}(a)x\xi_{f}, \xi_{f})$ . This is well-defined by the assumption (ii).
,,
We approximate by avector state from
, For each , fix
for all $k=1$ ,withorthonormal. It follows
witharepresentation
is ucp and the mapthat the map : $Marrow M_{m_{j}}$ defined byMoreover,we see that$M_{m_{j}}arrow
M$
is
CP:
defined byfor any $a\in A$
in satisfies
and $x\in M’$ . Therefore, we have$\mathrm{B}(\mathcal{H})$$\psi$$\tau$$M\subset \mathrm{B}(\mathcal{H})$

$\pi$

$\pi$

$\pi$

$\tau\psi$$\Omega$$(\mathrm{i}\mathrm{i})\Rightarrow(\mathrm{i}\mathrm{i}\mathrm{i})$$\theta=\rho\sigma$$A$$\mathrm{M}_{n}$$\sigma$$\rho$$\mathrm{i}\mathrm{d}_{M}$$\mathrm{M}_{n}$$\Omega$$\mathrm{p}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}-\sigma$$f_{1}$$\theta\in\Omega$$\ldots$$f_{n}$$a_{1}$$\epsilon$$\ldots$$|f_{k}(a_{k})-f_{k}(\theta(a_{k}))|<\epsilon$$\ldots$$(\pi_{f}, \mathcal{H}_{f}, \xi_{f})$$n$$x_{1}$$\ldots$$\omega$$\omega$$\mathit{3}t$$\omega$$\otimes \mathcal{H}_{f};\exists\eta_{1}$$\ldots$$| \omega(a_{k}\otimes x_{k})-\sum_{j=1}^{l}((a_{k}\otimes x_{k})\eta_{j}, \eta_{j})|<\epsilon$$\eta_{j}=\sum_{p=1}^{m_{\mathrm{j}}}\zeta_{j,p}\otimes\pi_{f}(b_{j,p})\xi_{f}$$\eta_{l}\in \mathcal{H}$$\ldots$$\otimes \mathrm{a}\pi f(M)\xi f$

$n$

$j$

$\{\zeta_{j,p}\}_{p=1}^{m_{\mathrm{j}}}$$\sigma_{j}(a)=[(a\zeta_{j,q}|\zeta_{j,p})]_{p,q}$$\sigma_{j}$$\rho_{j}([\alpha_{pq}]_{p,q})=\sum_{p,q}\alpha_{pq}b_{j,p}^{*}b_{j,q}$$\rho_{j}$$\Omega$$\theta=\sum_{j=1}^{l}\rho_{j}\sigma_{j}$$( \pi_{f}(\theta(a))x\xi_{f},\xi_{f})=\sum_{j=1}^{k}((a\otimes x)\eta_{j}, \eta_{j})$$|f_{k}(a_{k})-f_{k}( \theta(a_{k}))|=|\omega(a_{k}\otimes x_{k})-\sum_{j=1}^{k}((a_{k}\otimes x_{k})\eta_{j}, \eta_{j})|<\epsilon$

and we are done.
in Theorem 1.
. See the proof ofin Theorem 1.
. See the proof offor all $k=1$ ,$(\mathrm{i}\mathrm{i}\mathrm{i})\Rightarrow(\mathrm{i}\mathrm{i})$$(\mathrm{i}\mathrm{i})\Rightarrow(\mathrm{i})$$\ldots$,$n$$(\mathrm{i}\mathrm{i}\mathrm{i})\Rightarrow(\mathrm{i})$$(\mathrm{i})\Rightarrow(\mathrm{i}\mathrm{i})$$\square$

48
NARUTAKA OZAWA$(’\rfloor\backslash \Re \mathrm{g}\overline{\mathrm{f}\mathrm{f}})$. Astate on
Let be aC’-subalgebra in$a\in
A$ and any
if it satisfies $\varphi(ax)=\varphi(xa)$ for anyof Kirchberg [Ki3] generalizes Connes’ result [CO1] on$A$$\mathrm{B}(\mathcal{H})$$\mathrm{B}(\mathcal{H})$$\varphi$

is called ahypertrace for. The following theorem-factors.

$A$$x\in \mathrm{B}(\mathcal{H})$$\mathrm{I}\mathrm{I}_{1}$, TFAE.
Theorem 3. For a tracial state on a C’-subalgebra in.
(i). The trace extends to a hypertrace onsuch that
and(ii). There is a net of $ucp$ maps :
,
any
.
in
for
is continuous.
(iii). The functionalis(iv). The representationis the $GNS$-triplet for and the representation
continuous, where.
is given by:
for $a\in A$ and$A$$\tau$$\tau$$\mathrm{B}(\mathcal{H})$$\mathrm{B}(\mathcal{H})$$\varphi$$Aarrow \mathrm{M}_{n(i)}$$\theta_{i}$$\tau(a)=\lim_{i}\mathrm{t}\mathrm{r}_{n(:)}(\theta_{\dot{l}}(a))$$a$$\lim_{:}\mathrm{t}\mathrm{r}_{n(:)}(\theta:(ab^{*})-\theta_{i}(a)\theta:(b^{*}))=0$

$b$

$A$

$\sigma:A\otimes\min\overline{A}\ni\sum_{k}a_{k}\otimes\overline{b_{k}}\vdasharrow\tau(\sum_{k}a_{k}b_{k}^{*})\in \mathbb{C}$$\pi:A\otimes_{\min}\overline{A}\ni\sum_{k}a_{k}\otimes\overline{b_{k}}\vdasharrow\sum_{k}\pi_{\tau}(a_{k})\pi_{\tau}^{\mathrm{c}}(\overline{b_{k}})\in \mathrm{B}(\mathcal{H}_{\tau})$$(\pi_{\tau}, \mathcal{H}_{\tau}, \xi_{\tau})$

Proof.

$(\mathrm{i})\Rightarrow(\mathrm{i}\mathrm{i})$$\overline{b}\in\overline{A}$$\pi_{\tau}^{\mathrm{c}}(\overline{b})\pi_{\tau}(a)\xi_{\tau}=\pi_{\tau}(ab^{*})\xi_{\mathcal{T}}$$\overline{A}arrow \mathrm{B}(??)$$\pi_{\tau}^{\mathrm{c}}$$\tau$. The following proof is taken from [Ha2]. To prove (ii), we give,
. We approximate byourselves afinite set of unitaries in andclass operator on 7{ with $\mathrm{R}(h)=1$ . By astandard
where is apositive
approximation argument, we may assume that we have $|R(hu)$ $-\tau(u)|<\epsilon$ andfor $u\in E$ , and that is of finite rank and has no irrational, $p_{m}/q$,
be the non-zero eigenvalues of
eigenvalues; let,.
with the corresponding eigenvectors ,.
the -dimensional Hilbert space with adistinguished basis
We denote by$p=\max\{p_{1},\ldots,p_{m}\}$by.
and define isometries :Put. Identifying withbe the concatenation of
Finally let
by $\theta(a)=V^{*}(a\otimes 1)V$ . It follows that
, we define a ucp map, we havefor any $a\in A$ , and denoting
we have
$A$

$E$

$h$

$\epsilon>0$$\mathrm{b}(h\cdot$$\varphi$$)$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$$h$$||h-uhu^{*}||_{1,\mathrm{b}}<\epsilon$$p_{1}/q$$\zeta_{1}$$\ldots$$\mathbb{C}^{d}$$h$$(p_{1}, \ldots,p_{m}, q\in \mathrm{N})$$\ldots$$\zeta_{m}\in \mathcal{H}$$d$$\{\delta:\}_{\dot{l}=1}^{d}$$V_{k}$$\mathbb{C}^{pk}arrow \mathcal{H}\otimes \mathbb{C}^{p}$$V:\oplus_{k=1}^{m}\mathbb{C}^{pk}arrow \mathcal{H}\otimes \mathbb{C}^{p}$$V_{k}\delta_{i}=\zeta_{k}\otimes\delta:$$V_{k}’ \mathrm{s}$$\mathrm{M}_{q}$$\theta:Aarrow \mathrm{M}_{q}$$\mathrm{B}(\oplus_{k=1}^{m}\mathbb{C}^{pk})$$u_{k,l}=(u\zeta_{l}|\zeta_{k})$$\mathrm{t}\mathrm{r}_{q}(\theta(a))=\mathrm{T}\mathrm{r}(ha)$$\mathrm{t}\mathrm{r}_{q}(\theta(uu^{*})-\theta(u)\theta(u^{*}))=\sum_{k,1}|u_{k,l}|^{2}(p_{k}-\min\{p_{k},p_{l}\})/q$$\leq(\sum_{k,l}|u_{k,l}|^{2}(p_{k}^{1/2}+p_{l}^{1/2})^{2}/q)^{1/2}(\sum_{k,l}|u_{k,l}|^{2}(p_{k}^{1/2}-p_{l}^{1/2})^{2}/q)^{1/2}$$=||h^{1/2}u+uh^{1/2}||_{2,\mathrm{R}}||h^{1/2}u-uh^{1/2}||_{2,\mathrm{h}}$$\leq 2||hu-uh||_{1,\mathrm{R}}<2\epsilon$

for $u\in E$ , where we have used the Powers-Stormer inequality [PS] in the last line.
. The net of states : :is well-defined and converges to the functional .
. This immediately follows from the fact that is cyclic for.
and the corresponding vector state (which is ) is continuous onucpof
and letbe aextension. Let-bimodule map, we have that
. Since is an. It follows that the desired hypertrace extension
and
.
by
given
of is
$\sigma$$(\mathrm{i}\mathrm{i})\Rightarrow(\mathrm{i}\mathrm{i}\mathrm{i})$$A \otimes\min\overline{A}\ni\sum_{k}a_{k}\otimes\overline{b_{k}}|arrow \mathrm{t}\mathrm{r}_{n(i)}(\sum_{k}\theta:(a_{k})\theta:(b_{k}^{*}))\in \mathbb{C}$$\sigma$$\sigma$$(\mathrm{i}\mathrm{v})\Rightarrow(\mathrm{i})$$\Psi:\mathrm{B}(\mathcal{H})\otimes_{\min}\overline{A}arrow \mathrm{B}(\mathcal{H}_{\tau})$$x1arrow\Psi(x\otimes 1)\in \mathrm{B}(\mathcal{H}_{\tau})$$\psi|_{A}=\pi_{\tau}$$\tau$$\pi(A\otimes_{\mathrm{a}\mathrm{g}}\overline{A})$$\xi_{\tau}$$(\mathrm{i}\mathrm{i}\mathrm{i})\Rightarrow(\mathrm{i}\mathrm{v})$$\Psi$$\psi(\mathrm{B}(\mathcal{H}))\subset\pi_{\tau}(A)’$$\varphi(x)=(\psi(x)\xi_{\mathcal{T}}|\xi_{\tau})$$A \otimes\min\overline{A}$$\pi$$\psi:\mathrm{B}(\mathcal{H})\ni$$(A\otimes_{\min}\overline{A})$$\varphi$$\square$

49
ASURVEY

2. APOOR$\mathrm{M}\mathrm{A}\mathrm{N}’ \mathrm{S}$‘’$\mathrm{N}\mathrm{U}\mathrm{C}\mathrm{L}\mathrm{E}\mathrm{A}\mathrm{R}\mathrm{I}\mathrm{T}\mathrm{Y}\Leftrightarrow \mathrm{A}\mathrm{M}\mathrm{E}\mathrm{N}\mathrm{A}\mathrm{B}\mathrm{I}\mathrm{L}\mathrm{I}\mathrm{T}\mathrm{Y}$We prove aversion of
Let be a -algebra. The Haagerup norm for. (Sadly, it is still complicated.)is defined as$‘ \mathrm{n}\mathrm{u}\mathrm{c}\mathrm{l}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{y}\Leftrightarrow \mathrm{a}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{a}\mathrm{b}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}’$$A$$C^{*}$$T\in A\otimes_{\mathrm{a}}$$||T||_{h}= \inf\{||\sum a_{k}a_{k}^{*}||^{1/2}||\sum b_{k}^{*}b_{k}||^{1/2} :$A$

T=\sum_{k=1}^{n}a_{k}\otimes b_{k}\}$,andas a product ofRegardingiscontractive
we see that the product map, then we put
andw.r.t. the Haagerup norm. If.
by $\Phi_{T}(x)=\sum_{k}a_{k}xb_{k}$ . Again, it can be seen that:
We say aunital -algebraA is amenable $w.r.t$ . the Haagerup tensor product if there
satisfying thatinis anet,(i).
for all $i\in I$ ,(i).
for all $x\in A$ .
(iii).
It is clear that amenability w.r.t. the Haagerup tensor product is formally weaker
than the usual amenability. There are natural equivalent definitions of this concept
via virtual diagonal and via cohomology. Consult [Ru] for this matter.
Theorem 4. A C’-algebra is nuclear if and only if it is amenable $w.r.t$ . the Haagerup
tensor product.
Passing from this theorem to Theorem 1seems require aserious tool such as noncommutative Grothendieck inequality. Consult [Ef] and [Ru].
We first show the ‘if part. The following proof is taken from [BP]. Take afaithful
be as above. Then, theinand letnormal representation
is aquasi-expectation from B(??) onto $A’$ .
point-weak’ cluster point of the net
It follows that $A’$ , and afortiori $A’=A^{**}$ , is injective. See [BP] [Ru] for the proof.
For the proof of the ‘only if part, we need the following ingredients; atheoremof Kirchberg [Ki2] saying that aseparable nuclear -algebra is asubquotient of the
CAR-algebra (see [Wa2] for asimple proof), and Kasparov’s Stinespring theorem[Ka]. The following result, inspired from [KS], is sufficient for the ‘only if’ part.
Lemma 5. Let be a unital nuclear C’-algebra, $F$ be a finite set of unitaries in ,. Then, there arewhich are in the connected component of the identity and letsuch that $xx^{*}=1$ for $x\in G$ and for any
and a finite subset in$u\in F$ , there is a bijectionof onto with $||ux-f(x)||<\epsilon$ for $x\in G$ .
andHere, for $x=[x_{1}, \ldots, x_{n}]\in \mathrm{M}_{1,n}(A)$ and $u\in A$ , we define$[a_{1}, \ldots, a_{n}]\in \mathrm{M}_{1,n}(A)$$\sum a_{k}b_{k}$$[b_{1}, \ldots, b_{n}]^{T}\in \mathrm{M}_{n,1}(A)$$p:A \otimes_{\mathrm{a}\mathrm{g}}A\ni\sum_{k}a_{k}\otimes b_{k}\vdasharrow\sum_{k}a_{k}b_{k}\in A$$T= \sum_{k}a_{k}\otimes b_{k}\in A\otimes_{\mathrm{a}\mathrm{g}}A$$A\subset \mathrm{B}(\mathcal{H})$$||\Phi_{T}||_{\mathrm{c}\mathrm{b}}\leq||T||_{h}$$\Phi_{T}$$\mathrm{B}(\mathcal{H})arrow \mathrm{B}(\mathcal{H})$$C^{*}$$\{T_{\dot{l}}\}_{i\in I}$$A\otimes_{\mathrm{a}}\mathrm{g}$$A$$\sup_{:}||T_{\dot{l}}||_{h}<\infty$$\mathrm{p}(\mathrm{T}\mathrm{i})=1$$\lim_{i}||x\cdot T_{i}-T_{\dot{l}}\cdot x||_{h}=0$$\{T_{\dot{l}}\}:\in I$$A^{**}\subset \mathrm{B}(\mathcal{H})$$A\otimes_{\mathrm{a}\mathrm{g}}$$A$$\{\Phi_{T_{*}}.\}$$C^{*}$$A$$A$$\epsilon>0$$G$$n\in \mathrm{N}$$\mathrm{M}_{1,n}(A)$

$G$

$f$

$G$

$xx^{*}= \sum x_{k}x_{k}^{*}$$ux=[ux_{1}, \ldots, ux_{n}]\in \mathrm{M}_{1,n}(A)$.

is separable. By Kirchberg’s theorem [Ki2], there is
to some unitala ucp map from the CAR-algebra $B$ onto , whose restriction-homomorphism onto .
C’-subalgebra in $B$ becomes a

Proof. We may assume that

$A$

$A$

$\varphi|_{\tilde{A}}$$\varphi$$\tilde{A}$$A$$\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}*$50
NARUTAKA OZAWA$(/\rfloor\backslash \Re \mathrm{g}\overline{\mathrm{f}\mathrm{f}})$of unitaries in , which are in the connected. Lifting each element in , we find afinite
component of the identity, and, there is afinite set
isof unitaries in suchset of unitaries in . Sinceabijection
of
,
onto
with
any
is
for
there
that for. By Kasparov’s Stinespring theorem [Ka], there is aunital representation ofsuch thatfor $b\in B$ , whereon the Hilbert -module. We observe that
for
since
,
,
,
.
For
each
we
by
aunitary
defineandis
We give ourselves afinite set

$F$

$A$

$\epsilon>0$$B$$\tilde{A}$$\tilde{F}$

$F$

$B$

$\tilde{G}$$\mathrm{A}\mathrm{F}$$\tilde{u}\in\tilde{F}$$\tilde{G}$$\tilde{f}$$||\tilde{u}w-\tilde{f}(w)||<\epsilon/2$$\tilde{G}$$w\in\tilde{G}$

$\pi$

$A$

$B$

$\tilde{u}\in\tilde{F}$$\pi(\tilde{u}^{*})\zeta=(u^{*}, 0,0, \ldots)=\zeta u^{*}$$\zeta=(1,0,0, \ldots)\in \mathcal{H}_{A}$$w\in\tilde{G}$$||\pi(\tilde{u}^{*})||=1$$\varphi(\tilde{u}^{*})=u^{*}$$\pi(w^{*})\zeta=(w_{1}’, w_{2}’, \ldots)\in \mathcal{H}_{A}$$n\in \mathrm{N}$$\varphi(b)=\langle\zeta, \pi(b)\zeta\rangle$$\mathcal{H}_{A}=\ell_{2}\otimes A$. It follows thatto be large enough so that$w_{1}$$w_{2}$$\ldots\in A$,). Takefor alland put$G=\{\hat{w}:w\in
G\}\subset\mathrm{M}_{1,n}(A)$. The setandare what we desired.
0$\pi((\tilde{u}w)^{*})\zeta=((uw_{1})^{*}, (uw_{2})^{*}$$\ldots$$w\in\tilde{G}$$|| \sum_{k\geq n}w_{k}w_{k}^{*}||<\epsilon/8$$\hat{w}=$$[w_{1}, \ldots, w_{n-1}, (\sum_{k\geq n}w_{k}w_{k}’)^{1/2}]\in \mathrm{M}_{1,n}(A)$the bijection

$f$

on

$G$

induced from

3. ANON-OPERATOR$\tilde{f}$$\mathrm{A}\mathrm{L}\mathrm{G}\mathrm{E}\mathrm{B}\mathrm{R}\mathrm{A}\mathrm{I}\mathrm{S}\mathrm{T}’ \mathrm{S}$NON-AMENABILITY OF$\mathrm{B}(\ell_{2})$(or any von Neumann algebra which
We present here aproof of the fact thatis not subhomogeneous) is not amenable. This proof was suggested by G. Pisier.
$\mathrm{B}(\ell_{2})$Theorem 6. The Banach algebra$\mathrm{B}(\ell_{2})$is not amenable.

insatisfying theActually we will show that there is no net
Intheory,operator
2.
ofalgebra
(iii)
stead
(ii)
in
Sectionwe needandconditionand operator
the following ingredients; Kazhdan’s property (T) for, say,(T)
property
said
to
have
Kazhdan’s
if for any
group
isinequalities. Adiscrete$\kappa>0$$E$
and adecreasingof generators in , there are aconstantfinite subset$\{T_{i}\}_{i}$$\mathrm{B}(\ell_{2})\otimes_{\mathrm{a}\mathrm{g}}\mathrm{B}(\ell_{2})$$SL(3, \mathbb{Z})$$\Gamma$$\Gamma$such that the following is true: if
withandis aunit vector with
is aunitary representation on aHilbert space$\epsilon=\max_{s\in E}||\pi(s)\xi-\xi||<\kappa$, then there is aunit vector $\eta\in?t$ with $||\xi-\eta||<f(\epsilon)$. It is well-known that the grouphassuch that $\pi(s)\eta=\eta$ for all
Kazhdan’s property (T). We refer the reader to [HV] for the information of Kazhdan’sclass operators and on aHilbert space, the Powersproperty (T). For any
inequality [PS] says that $||h^{1/2}-k^{1/2}||_{2}^{2}\leq||h-k||_{1}$ and Kosaki’s inequality[Ko] says that $|||h|-|k|||_{1}\leq(2||h+k||_{1}||h-k||_{1})^{1/2}$ . The proof of these inequalities
for matrices (which we will need) are rather elementary (cf. [Me]).

function$\lim_{\epsilonarrow 0}f(\epsilon)=0$$f:\mathbb{R}_{+}arrow \mathbb{R}_{+}$$\pi$$\xi\in \mathcal{H}$$\mathcal{H}$$s\in\Gamma$$SL(3, \mathbb{Z})$$h$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$$k$$\mathrm{S}\mathrm{t}\emptyset \mathrm{r}\mathrm{m}\mathrm{e}\mathrm{r}$Lemma 7. Let $E$ be a finite set of generators of . Then, there are a constant $\delta>0$
which satisfy theand a decreasing function $c:(0, \delta)arrow(0,1)$ withis an irreducible representation andfollowing property. If$\Gamma$$\lim_{\epsilonarrow 0}c(\epsilon)=0$$\pi:\Gammaarrow \mathrm{M}_{n}$$T= \sum_{=i1}^{r}a_{i}\otimes b_{\dot{l}}\in \mathrm{M}_{n,\infty}\otimes \mathrm{M}_{\infty,n}$

is such that$\sum_{i=1}^{r}a_{i}b:=I_{n}$$r\geq(1-c(\epsilon))n$.

and$\epsilon$$= \max_{s\in E}||\pi(s)\cdot$ $T-T\cdot$ $\pi(s)||_{h}<\delta$, then we have

51
ASURVEY

be the correspondinganda\^i $i=I_{n}$ , we have
. Sinceaiekhorthogonal projections. Let. On the other hand, we claim that
that

Proof. Let$\{\delta_{k}\}_{k=1}^{\infty}$

be an orthonormal basis of$T(k)= \sum_{i=1}^{r}$$\ell_{2}$$\{e_{k}\}$$\in \mathrm{M}_{n}$$\sum_{i=1}^{r}$$\sum_{k=1}^{\infty}||T(k)||_{1’\mathrm{R}},\geq n$$\forall s\in E$$\sum_{k=1}^{\infty}||\pi(s)T(k)\pi(s)^{*}-T(k)||_{1,\mathrm{R}}<\epsilon n$.

Indeed, this follows from astandard homogeneity trick and the following inequality;, we haveandfor$y\in \mathrm{M}_{\infty,n}$$x\in \mathrm{M}_{n,\infty}$$\sum_{k=1}^{\infty}||xe_{k}y||_{1,\mathrm{R}}=\sum_{k=1}^{\infty}||x\delta_{k}||_{\ell_{2}}||y^{*}\delta_{k}||_{\ell_{2}}$$\leq\frac{1}{2}\sum_{k=1}^{\infty}(||x\delta_{k}||_{\ell_{2}}^{2}+||y^{*}\delta_{k}||_{\ell_{2}}^{2})=\frac{1}{2}\mathrm{T}\mathrm{r}_{n}(xx^{*}+y^{*}y)$These inequalities implies the existence of$\forall s\in E$

We put$k_{0}\in \mathrm{N}$such that$||\pi(s)T(k_{0})\pi(s)^{*}-T(p_{0})||_{1’\mathrm{R}},<|E|\epsilon||T(k_{0})||_{1,\mathrm{R}}$$h=T(k_{0})/||T(k_{0})||_{1,\mathrm{T}\mathrm{r}}$.

.

. It follows from Kosaki’s inequality that$||\pi(s)|h|\pi(s)^{*}-|h|||_{1’\mathrm{R}},\leq(2||\pi(s)h\pi(s)^{*}+h||_{1,\mathrm{b}}||\pi(s)h\pi(s)^{*}-h||_{1^{\mathrm{r}}\mathrm{R}},)^{1/2}<(4|E|\epsilon)^{1/2}$.

Combined with Powers-Stormer inequality, this implies$\forall s\in E$

If
(here

$\epsilon>0$$f$$||\pi(s)|h|^{1/2}\pi(s)^{*}-|h|^{1/2}||_{2,\mathrm{R}}<(4|E|\epsilon)^{1/4}$.

is small enough, then it follows Schur’s lemma that for $c(\epsilon)=f((4|E|\epsilon)^{1/4})$
is as in the above definition of Kazhdan’s property (T)), we have$|||h|^{1/2}-n^{-1/2}I_{n}||_{2,?\mathrm{Y}}^{2}<c(\epsilon)$

since fixed vectors for the representation Ad
amultiple of identity. Sincecompletes the proof.

$\pi$

on the Hilbert-Schmidt class is$h\leq r$ , we have $r\geq(1-c(\epsilon))n$ . Thisof$\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}|h|^{1/2}=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}$$S_{2}$$\Gamma$$\square$: $SL(3, Z)arrow \mathrm{M}_{n(k)}$ be asequence of finite dimensionalas $karrow\infty$ . We identify with
irreducible representations such that

Proof of Theorem 6. Let$\pi_{k}$$n(k)arrow \mathrm{o}\mathrm{o}$$\ell_{2}$$\bigoplus_{\oplus_{s=1}^{k1}\infty}\infty\ell_{2}^{n(k)}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{o}\mathrm{t}\mathrm{e}\mathrm{b}\mathrm{y}=P_{k}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{h}\mathrm{o}\mathrm{g}\mathrm{o}\mathrm{n}\mathrm{a}1\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{f}\mathrm{r}\mathrm{o}\mathrm{m}\ell_{2}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{o}\ell_{2}^{n(k}\mathrm{L}\mathrm{e}\mathrm{t}\pi(s)=\pi_{n}(s)\in \mathrm{B}(\ell_{2})\mathrm{f}\mathrm{o}\mathrm{r}s\in SL(3\mathbb{Z})\mathrm{a}\mathrm{n}\mathrm{d}1\mathrm{e}\mathrm{t}\delta>0\mathrm{b}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{L}\mathrm{e}\mathrm{m}\mathrm{m}\mathrm{a}7.\mathrm{T}\mathrm{o}\mathrm{s}\mathrm{h}\mathrm{o}\mathrm{w}\mathrm{B}(\ell_{2})$is not amenable by reductio ad absurdum, suppose that there is.
and $= \max_{s\in E}||\pi(s)\cdot$ $T-T\cdot$
such that$r>(1-c(\epsilon))n(k)$, we obtain
Applying Lemma 7to.
isabsurd.
for all This
$T= \sum_{\dot{l}=1}^{r}a:\otimes b_{:}\in$$\mathrm{B}(\ell_{2})\otimes \mathrm{B}(\ell_{2})$$\sum_{\dot{l}}^{r}=1a_{i}b_{i}=I_{\ell_{2}}$$\epsilon$$\pi(s)||_{h}<\delta$$P_{k}\cdot T\cdot P_{k}\in \mathrm{M}_{n(k),\infty}\otimes \mathrm{M}_{\infty,n(k)}$$k$$\square$

52
NARUTAKA OZAWA$(’\rfloor\backslash \mathrm{R}\mathrm{g}\ovalbox{\tt\small REJECT})$

4. EXACTNESS
AC’-algebra$A$

is said to be exact if$\mathrm{O}arrow A\otimes_{\min}Jarrow A\otimes_{\min}Barrow A\otimes_{\min}B/Jarrow \mathrm{O}$is exact for any C’-algebra and any closed 2-sided ideal in . As we will seein the proof of Theorem 8, it suffices to check exactness of the above sequence only
for eitheror. Kirchberg [Ki2]showed that exactness is characterized by the following property, known as nuclear
embeddability.
$B$

$J$

$J= \oplus_{n=1}^{\infty}\mathrm{M}_{n}\triangleleft B=\prod_{n=1}^{\infty}\mathrm{M}_{n}$$B$$J=\mathrm{K}(\ell_{2})\triangleleft B=\mathrm{B}(\ell_{2})$Theorem 8. Let
be an exact C’-algebra and let
be an increasing
sequence of projections on , which converges strongly to the identity on 7#. We:
denote by
the compression. Then, there es a net of $ucp$ maps$\lim_{i}||\theta_{i}\varphi_{n(i)}(a)-a||=0$:
such thatfor all $a\in A$ .
Proof. The following proof is taken from [Pi]. Given afinite dimensional subspace$E\subset A$ and$>0$ , we will findand a ucp map
such that. It is easy to see thatis one-t0-0ne for large enough. We:
claim that
. Since factors throughwhen
is monotonically decreasing and the limit $c\geq 1$$m>n$ , the sequence
exists. For each , we take $x_{n}\in E\otimes M(H)$ with $||x_{n}||=1$ and$c^{-1}+n^{-1}$ , and let $x=(x_{n})_{n=1}^{\infty} \in\prod_{n=1}^{\infty}(E\otimes \mathrm{M}_{k(n)})=E\otimes M\subset A\otimes M$ ,whereis the-direct product. (Thatis$E$ isdimensional.)Letbecausefinitebe the -direct product,
which is an ideal in $M$ . Since $||x_{n}||=1$ for all , we have $||x+A\otimes J||_{A\otimes M/A\otimes J}=1$ .
On the other hand, if $M/J$ is faithfully represented on , then we have
$A\subset \mathrm{B}(\mathcal{H})$$(P_{n})_{n=1}^{\infty}$$\mathcal{H}$$\varphi_{n}$$\theta_{i}$$\mathrm{B}(P_{n(:)}\mathcal{H})$$\mathrm{B}(\mathcal{H})arrow \mathrm{B}(P_{n}\mathcal{H})$$arrow \mathrm{B}(\mathcal{H})$$n\in \mathrm{N}$$\epsilon$$\theta:\mathrm{B}(P_{n}\mathcal{H})arrow \mathrm{B}(\mathcal{H})$$\varphi_{n}|_{E}$$||\theta\varphi_{n}|_{E}-\mathrm{i}\mathrm{d}_{E}||_{\mathrm{c}\mathrm{b}}<\epsilon$$\lim_{narrow\infty}||(\varphi_{n}|_{E})^{-1}$$\varphi_{n}(E)arrow E||_{\mathrm{c}\mathrm{b}}=1$$n$$\varphi_{n}$$\varphi_{m}$$||(\varphi_{n}|_{E})^{-1}||_{\mathrm{c}\mathrm{b}}$$n$$||(\varphi_{n}\otimes \mathrm{i}\mathrm{d}_{k(n)})(x_{n})||\leq$$M= \prod_{n=1}^{\infty}\mathrm{M}_{k(n)}$$\prod_{n=1}^{\infty}(E\otimes \mathrm{M}_{k(n)})=E\otimes M$$\ell_{\infty}$$J=\oplus_{n=1}^{\infty}\mathrm{M}_{k(n)}$$c_{0}$$n$$\mathcal{K}$$||(\mathrm{i}\mathrm{d}_{A}\otimes Q)(x)||_{A\otimes M/J}=||(\mathrm{i}\mathrm{d}_{A}\otimes Q)(x)||_{\mathrm{B}(\mathcal{H}\otimes \mathcal{K})}$$= \lim_{narrow\infty}||(\varphi_{n}\otimes \mathrm{i}\mathrm{d}_{M/J})(\mathrm{i}\mathrm{d}_{A}\otimes Q)(x)||_{\mathrm{B}(P_{n}\mathcal{H}\otimes \mathcal{K})}$$= \lim_{narrow\infty}||(\mathrm{i}\mathrm{d}_{\mathrm{B}(P_{n}\mathcal{H})}\otimes Q)((\varphi_{n}\otimes \mathrm{i}\mathrm{d}_{M})(x))||_{\mathrm{B}(P_{n}\mathcal{H}\Theta \mathcal{K})}$$\leq\lim_{narrow\infty}\lim_{marrow}\sup_{\infty}||(\varphi_{n}\otimes \mathrm{i}\mathrm{d}_{\mathrm{M}_{k(m)}})(x_{m})||_{\mathrm{B}(P_{n}\mathcal{H})\otimes \mathrm{M}_{k(m)}}$$\leq\lim_{marrow}\sup_{\infty}||(\varphi_{m}\otimes \mathrm{i}\mathrm{d}_{\mathrm{M}_{k(m)}})(x_{m})||_{\mathrm{B}(P_{m}\mathcal{H})\otimes \mathrm{M}_{k(m)}}$$\leq c^{-1}$Hence, by the assumption on exactness, we have $1\leq c^{-1}$ and obtain the claim.

for sufficiently large . Using the injectivity
It follows that
, we extend
to aself-adjoint map $:ofwith thesame -norm. Since is unital self-adjoint, we find awithby Lemma 9below. This completes the proof.
Cl
Lemma 9(Kirchberg). If $:is a unital self-adjoint map, then there isa $ucp$ map
such that.
$||(\varphi_{n}|_{E})^{-1}||_{\mathrm{c}\mathrm{b}}\leq 1+\epsilon$$(\varphi_{n}|_{E})^{-1}$$\mathrm{B}(\mathcal{H})$$\mathrm{c}\mathrm{b}$$n$$\mathrm{B}(P_{n}\mathcal{H})arrow \mathrm{B}(\mathcal{H})$$\psi$$\mathrm{u}\mathrm{c}\mathrm{p}\theta:\mathrm{B}(P_{n}\mathcal{H})arrow \mathrm{B}(\mathcal{H})$$||\theta-\psi||_{\mathrm{c}\mathrm{b}}<\epsilon$$Aarrow \mathrm{B}(\mathcal{H})$$\theta:Aarrow \mathrm{B}(\mathcal{H})$$||\theta-\psi||_{\mathrm{c}\mathrm{b}}\leq||\psi||_{\mathrm{c}\mathrm{b}}-1$

53
ASURVEY

Proof. Use Wittstock decomposition theorem or Stinespring theorem for cb maps, or
consult Wassermann’s monograph [Wa2].
$\square$

The converse of the above theorem due to Wassermann states that
Theorem 10. Letof finite rank maps

$A\subset \mathrm{B}(\mathcal{H})$$\varphi_{n}$:be a separable
such that-algebra such that there is a sequence$Aarrow \mathrm{B}(??)$$\lim_{n}||(\varphi_{n}\otimes \mathrm{i}\mathrm{d}_{\mathrm{B}(\ell_{2})})(x)-x||_{\mathrm{B}(\mathcal{H})\otimes \mathrm{B}(\ell_{2})}=0$

for anythat$x\in A\otimes \mathrm{B}(\ell_{2})$, ($e.g$ . a sequence of finite rank $ucp$ maps
then is exact.

$\varphi_{n}$:$Aarrow \mathrm{B}(\mathcal{H})$

such

$A$

$\forall a\in A\lim_{n}||\varphi_{n}(a)-a||=0)$

Proof. We give ourselves aC’-algebra , an ideal. It follows that
may assume that$B$

$J\triangleleft B$

and$x\in \mathrm{k}\mathrm{e}\mathrm{r}(\mathrm{i}\mathrm{d}_{A}\otimes Q)$. We$B\subset \mathrm{B}(\ell_{2})$.

$(\mathrm{i}\mathrm{d}_{\mathrm{B}(\mathcal{H})}\otimes Q)(\varphi_{n}\otimes \mathrm{i}\mathrm{d}_{\mathrm{B}(\ell_{2})})(x)=(\varphi_{n}\otimes \mathrm{i}\mathrm{d}_{\mathrm{B}(\ell_{2})})(\mathrm{i}\mathrm{d}_{A}\otimes Q)(x)=0$

Since$\varphi_{n}$. Taking limit, we haveis of finite rank, we have
. This completes the proof.

$(\varphi_{n}\otimes \mathrm{i}\mathrm{d}_{\mathrm{B}(\ell_{2})})(x)\in A\otimes J$$x\in A\otimes J$$\square$

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DEPARTMENT OF MATHEMATICAL SCIENCE, UNIVERSITY OF TOkyO, 153-8914, JAPAN-mail address: narutakaQms. -tokyo.

$E$$\mathrm{u}$$\mathrm{a}\mathrm{c}.\mathrm{j}\mathrm{p}$