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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

Citations:

Zbl 0168.276
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References:

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