1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

12.3. Weak amenability 365 Corollary 12.3.7. Let I'= YIA be the wreath product ofY by A. If JYJ > 1 and A is nonamenable, the ...

366 12. Approximation Properties for Groups Though the context should always be clear, one must be careful not to mix up Acb for ...

12.3. Weak amenability be an isometry such that, for each s Er, 1 VFb"s = /IDT I: 88 -lp 0 b"p 0 b"cr(p)-lscr(s-lp)· v IFI pEF 3 ...

368 12. Approximation Properties for Groups Proof. It is not hard to see that D = {a~ f : llall < 1, llfll < 1} is a conve ...

12.4. Another approximation property 369 for C*-algebras Ai and A2. The inequality Acb(Ai@A2)::; Acb(Ai)Acb(A2) is trivial. To p ...

370 12. Approximation Properties for Groups Remark 12.4.2. A Banach space Xis said to have the AP (approximation property) if th ...

12.4. Another approximation property 371 Proof. We only prove assertion (2). Denote by F(A) the set of all con- tinuous finite-r ...

372 12. Approximation Properties for Groups and Kirchberg [101] independently proved that an exact (or locally reflexive) C* -al ...

12.4. Another approximation property 373 By Lemma D. 7, if <{Ji E qr] converges to 1 weak*, then m'Pi © idlB(£2) -t idc~(r)Gl ...

374 12. Approximation Properties for Groups Exercise 12.4.5. Let r = I'1 * I'2 be the free product of I'1 and r2. Prove that r h ...

Weak Expectation Property and Local Lifting Property Chapter 13 This chapter contains a potpourri of concepts and results which ...

376 13. WEP and LLP A unital C*-algebra A has the lifting property (LP) (resp. local lifting prop- erty (LLP)) if any c.c.p. map ...

13.1. The local lifting property 377 between their multiplier algebras. Hence, by the first part of this proof, there is a u.c.p ...

378 13. WEP and LLP 13.2. Tensorial characterizations of the LLP and WEP To get tensorial characterizations of the LLP and WEP, ...

13.2. Tensorial characterizations of the LLP and WEP 379 Lemma 13.2.3. Let Xi c lBl(Hi) (i = 1, 2) be unital operator subspaces ...

380 13. WEP and LLP (2) A@maxlffi(.€^2 ) = A@lIB(.€^2 ) canonically if and only if A has the LLP; (3) C*(lF 00 ) @max B = C*(JF ...

13.3. The QWEP conjecture 381 We will only prove the equivalence of (1), (2) and (3); see [134] for the last condition (and the ...

382 13. WEP and LLP Lemma 13.3.5. Let A and B be unital C* -algebras. If B has the WEP and there exists a u.c.p. map cp: B ---+A ...

13.3. Tb.e QWEP conjecture 383 Since A has the LP, there is a c.c.p. lifting 'ljJ: A--"* B. Let A** c IIB('H) be a universal rep ...

384 13. WEP and LLP Of course, if C~(r) is QWEP, then so is L(r), by Lemma 13.3.6. But typically it's very hard to show reduced ...