## Discrete groups with Kazhdan’s property $$T$$ and factorization property are residually finite.(English)Zbl 0805.46066

Let $$G$$ be a discrete group. $$G$$ has factorization property (property (F)) if the two-sided regular representation $$\lambda\otimes \rho: G\times G\to {\mathcal L}(\ell_ 2(G))$$ given by $$\lambda\otimes \rho(g,h)= \lambda(g)\rho(h)$$ extends to a continuous *-representation from $$C^*(G)\otimes^{\min} C^*(G)$$ into $${\mathcal L}(\ell_ 2(G))$$ [cf. the author, Invent. Math. 112, 449-489 (1993), sec. 7].
Let $$G$$ be a discrete group with property $$T$$ of D. A. Kazhdan [Funct. Anal. Appl. 1, 63-65 (1967; Zbl 0168.276)]. We show that $$G$$ is maximally almost periodic if and only if $$G$$ has property (F). Since for discrete groups with property $$T$$, maximal almost periodicity is equivalent to residual finiteness, it follows that property (F) is not satisfied for all discrete groups.
In particular, $$G$$ is residually finite if it has property $$T$$ and has a faithful representation into a locally compact group $$H$$ such that $$C^*(H)$$ is nuclear (e.g. if $$H$$ is a connected Lie group) or if $$G$$ has property $$T$$ and a faithful unitary representation into the hyperfinite $$\text{II}_ 1$$ factor.

### MSC:

 46L55 Noncommutative dynamical systems 22D25 $$C^*$$-algebras and $$W^*$$-algebras in relation to group representations 46L35 Classifications of $$C^*$$-algebras

Zbl 0168.276
Full Text:

### References:

 [1] Alperin, R.: An elementary account of Selberg’s lemma. Enseign. Math.33, 269-273 (1987) · Zbl 0639.20030 [2] Arveson, W.: Notes on extensions of C*-algebras. Duke Math. J.44, 329-355 (1977) · Zbl 0368.46052 [3] Bernstein, I.N.: All reductive p-adic groups are tame. Funct. Anal. Appl.8, 91-93 (1974) · Zbl 0298.43013 [4] Choi, M.D.: A Schwarz inequality for positive linear maps on C*-algebras, Ill. J. Math.18, 565-574 (1974) · Zbl 0293.46043 [5] Connes, A.: Classification of injective factors. Ann. Math.104, 73-116 (1976) · Zbl 0343.46042 [6] Delzant, T.: Sous-groupes distingu?s et quotient des groupes hyperboliques. Preprint, Strasbourg 1991 · Zbl 0845.20027 [7] Gromov, M.: Hyperbolic groups. In: Gersten, S.M. (ed.) Essays in group theory (M.S.R.I. Publ. 8, pp. 75-263) Springer 1987 · Zbl 0634.20015 [8] Grothendieck, A.: Produits tensoriels topologiques et espaces nucl?aires M?m. Am. Math. Soc. 16 (1955) · Zbl 0072.12003 [9] de la Harpe, P., Valette, A.: La propri?t?T de Kazhdan pour les groupes localement compacts. Ast?risque 175 (1991) · Zbl 0759.22001 [10] de la Harpe, P., Robertson, A.G., Valette, A.: On exactness of group C*-algebras. Preprint 1991 · Zbl 0830.22004 [11] Kazhdan, D.: Connection of the dual space of a group with the structure of its closed subgroups. Funct. Anal. Appl.1, 63-65 (1967) · Zbl 0168.27602 [12] Kirchberg, E.: Positive maps and C*-nuclear algebras. In: Proc. Inter. Conference on Operator Algebras, Ideals and their Applications in Theoretical Physics, Leipzig, 1977 (pp. 255-257) Leipzig, Teubner 1978. [13] Kirchberg, E.: On non-semisplit extensions, tensor products and exactness of group C*-algebras. Invent. Math.112, 449-489 (1993). · Zbl 0803.46071 [14] Paterson, A.L.T.: The class of locally compact groupsG for whichC * (G) is amenable. In: Eymard, P. (ed.) Harmonic Analysis. (Lect. Notes Math., vol. 1359, pp. 226, 237). Berlin Heidelberg New York: Springer (1988) [15] Renault, J., Skandalis, G.: Lecture held at C*-conference in Oberwolfach, October 1991 [16] Robertson, A.G.: Property (T) forII 1 factors and unitary representations of Kazhdan groups. Preprint 1992, University of Newcastle, Australia [17] Wassermann, W.: C*-algebras associated with groups with Kazhdan’s propertyT. Ann. Math.99, 423-432 (1991) · Zbl 0754.46040
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.