374 12. Approximation Properties for GroupsExercise 12.4.5. Let r = I'1 * I'2 be the free product of I'1 and r2. Prove
that r has the AP if each ri has the AP. (Hint: Prove that, for every
i E {1,2} and d,n EN, there is a bounded map T: B 2 (I'i)--+ B2(r) such
that T(cp)(s) = On,mOi,id cp(sa) for s = s1 ... Sm with Sj E rij and ij # ij+i.)
12.5. References
(Relative) property (T) was introduced by Kazhdan (and Margulis). Since
then, several groups have been shown to have property (T) using formidable
machinery from Lie group theory or spectral geometry. The first "elementary
proof" of the fact that some infinite group (e.g., SL(3, Z)) has property (T)
was recently given by Shalom [174]. Theorem 12.1.7 (especially (1) =? (2))
is due to Jolissaint [91]. Lemma 12.1.8 is due to de la Harpe, Robertson
and Valette [83]. Theorem 12.1.10 is due to Margulis, but its quantitative
proof is due to Shalom [173]. Remark 12.1.11 is due to Burger [32]. The
proof of Theorem 12.1.14 is based on Shalom's work [174]. Theorem 12.1.15
is due to Zuk [201]. The notion of property (T) for van Neumann algebras
was introduced by Cannes and Jones [45]. Theorem 12.1.18 is a routine
modification of [45] to the relative property (T) context. We should mention
that an analogue of Theorem 12.1.7 is true for van Neumann algebras [144].
Theorem 12.1.19 is due to Cannes [43]. For a comprehensive treatment of
property (T), we refer to the book of Bekka, de la Harpe and Valette [15].
Haagerup's property was introduced by, well, Haagerup; Theorems 12.2.5
and 12.2.4 are proved in his seminal paper [75]. Theorem 12.2.9 comes from
[33]. Theorem 12.2.11 is due to de Cornulier, Stalder and Valette [55].
Theorem 12.2.15 was proved by Choda [34] (see also [90]), while Theo-
rem 12.2.16 is due to Popa [160]. For a comprehensive treatment of the
Haagerup property, we refer to the book of Cherix, Cowling, Jolissaint, Julg
and Valette [33].
The CBAP was also introduced by Haagerup. Theorem 12.3.3 and
its corollary were proved by Bozejko and Picardello [25]. The fact that
Acb(IFn) = 1 is due to De Canniere and Haagerup [54]. More general ver-
sions of Theorem 12.3.6 and its corollary were proved by Ozawa and Popa
[138]. Theorem 12.3.8 combines work of De Canniere and Haagerup [54],
Cowling [46], Haagerup [78] and Cowling and Haagerup [47]. See also [56]
for a proof of nonweak amenability of Z^2 ><1 SL(2, Z) and other groups. The-
orem 12.3.10 is due to Haagerup [78], while Theorem 12.3.13 was proved by
Cowling and Haagerup [47] for group algebras and by Sinclair and Smith
[1 75] in general.
Theorem 12.4.4 is due to Kraus [111]. Corollary 12.4.6 is work of Kirch-
berg [101]. Finally, Theorem 12.4.9 and Proposition 12.4.10 were proved by
Haagerup and Kraus [80].