1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

12.1. Kazhdan's property (T) 345 We identify the Pontryagin dual of Z with 'JI' = IR/Z by the pairing Z x 'JI' 3 (n, t) 1-+ e^2 ...

346 12. Approximation Properties for Groups Therefore, §2^1 (B1 U B2) c Bo U Bi, §1(B3 U B4) c Bo U B3, §2(B3 U B4) c Bo U B4. μ ...

12.1. Kazhdan's property (T) 347 Proof. Let s ~ [: ; :J E SL(3,Z) be given. By the Chinese Remainder Theorem, there exists m E Z ...

348 12. Approximation Properties for Groups for any unitary representation (7r, J-t) of I' and e E H, where Pi is the orthogonal ...

12.1. Kazhdan's property (T) 349 Set So = ~ 2-k7r(sk)S7r(sk)* E Co for an enumeration r = { sk}~ 1 and take a sequence ((n) of u ...

350 12. Approximation Properties for Groups s ES, and we set lvl = 2:: v(s) = IEI. Let L^2 (S, v) be the weighted L^2 -space on ...

12.1. Kazhdan's property (T) 351 Unfortunately, providing examples to which this theorem applies is not so easy -though there ar ...

352 12. Approximation Properties for Groups for every x E J. We define a T-preserving u.c.p. map c.p: M ---+ M by (c.p(a)x,fj) = ...

12.1. Kazhdan.'s property (T) 353 For a II1-factor M, we let Aut(M) denote the group of *-automorphisms of M, Int(M) c Aut(M) de ...

354 12. Approximation Properties for Groups Exercise l~!.1.3. Let I' be an ICC^7 group and let x: I'----? {z: lzl = 1} be a nont ...

12.2. The Haagerup property 355 ll - 'Pn(sk)I < 2-n for every k and n 2:: k. By the GNS construction (see Theorem 2.5.11), fo ...

356 12. Approximation Properties for Groups sto , ~t) b( so ~ 0 Figure 1. b(st) = b(s) +7r(s)b(t) In summary, we obtain the foll ...

12.2. The Haagerup property 357 o and x. We leave it to the reader to check w(x,y) = li((y) - ((x)JJ^2. Hence w is conditionally ...

358 12. Approximation Properties for Groups s EA) and the unit. We have to show that the set {xs Er: wr(xs) s N} is finite for e ...

12.2. The Haagerup property 359 Observe that the semidirect product Z^2 ><l SL(2, Z) of Z^2 by SL(2, Z) does not have the ...

360 12. Approximation Properties for Groups In recent years Popa has exploited the Haagerup property in combination with relativ ...

12.3. Weak amenability 361 and 0 is not contained in the ultraweakly closed convex hull of { wdw* : w E B unitary}. By Theorem F ...

362 12. Approximation Properties for Groups starts at x and eventually flows into w. It is not hard to see (cf. Figure 2) that n ...

12.3. Weak amenability 363 Proof. For simplicity, we assume that T is uniformly locally finite. As in the proof of Theorem 12.3. ...

364 12. Approximation Properties for Groups Let A <1 r be a normal subgroup. Then, r acts on A by conjugation: Ad(s)(a) = sas ...