A property (T) for C*-algebras

Details A property (T) for C*-algebras A property (T) for C*-algebras

2005-05-10

arxiv.org/abs/math/0505189v1

Mathematics

Operator Algebras

14 pages

Scientific paper

We define a notion of Property (T) for an arbitrary $C^*$-algebra $A$ admitting a tracial state. We extend this to a notion of Property (T) for the pair $(A,B),$ where $B$ is a $C^*$-subalgebra of $A.$ Let $\Gamma$ be a discrete group and $C^*_r(\Gamma)$ its reduced algebra. We show that $C^*_r(\Gamma)$ has Property (T) if and only if the group $\Gamma$ has Property (T) . More generally, given a subgroup $\Lambda$ of $\Gamma$, the pair $(C^*_r(\Gamma),C^*_r(\Lambda))$ has Property (T) if and only if the pair of groups $(\Gamma, \Lambda)$ has Property (T).

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