1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

388 13. WEP and LLP

tensor product):

0 ~ JK ®max F ~ B ®max F ~ CA ®max F -----?>- 0

l l l

0 __ _,,_ lK ® F -----,,.. B ® F __ .,,..CA ®max F ~ 0.

Since lK is nuclear, we have B ®max F = B ® F by the 5 Lemma. D

Remark 13.4.6. With additional effort, one can find B with the property

that B ®max B^0 P = B ® B^0 P. We sketch the argument. In the proof of

Lemma 13.4.5, we have

limsup 11(',Pn ® 'lj;~P)(y)ll = llY +(CJ® CF^0 P +CF® CJ^0 P)ll

n---+oo

for every y E CF® CF^0 P. Hence, we can take '11 in Lemma 13.4.5 so that

it satisfies

CF@CF^0 P ~ M@M^0 P

CJ® C F^0 P + CF ® C J^0 P K ® M^0 P + M ® K^0 P'

where M = IJMn(C) and K = ffiMn(C). The rest of the proof is similar

to that of Theorem 13.4.1.

13.5. Norms on JIB(£^2 ) 8 JIB(£^2 )

We close this chapter with another fundamental result.

Theorem 13.5.1 (Junge-Pisier). The C*-algebra IIB(.£^2 ) does not have the

LLP; in other words,

IIB(.£^2 ) ®max IIB(.£^2 ) # IIB(.£^2 ) ® JIB(.£^2 ).

For a Hilbert space H, we denote by H its complex conjugate Hilbert
space. More precisely, H 3 e H ~ E H is an isomorphism as a real Hilbert
space, but we have -Ae = )..~and (~, r;)?i = (e, TJ)?i for .A E C and e, TJ E H.
For x E JIB(H), we denote by x the corresponding operator in JIB(H). Finally,
the complex conjugate of a C -algebra A c IIB(H) is defined as A = { x E
IIB(H) : x EA}. We note that A is -isomorphic to the opposite A^0 P via the
adjoint map. The following lemma is well known, so we omit the proof.

Lemma 13.5.2. The Hilbert space H ® /(, can be identified with the space
S2(1C, H) of Hilbert-Schmidt class operators from/(, into H via the isomor-
phism e: H ® K -----+ S 2 ( K, H) given by

e(L: ei 0 rJi)( =I:(, T/i)x:ei