Discrete groups with Kazhdan's property \(T\) and factorization property are residually finite
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 \textit{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.
discrete group with property \(T\) of Kazhdan
scientific article; zbMATH DE number 683113