## 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
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### References:

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