Diagonal type conditions on group C*-algebras

Spronk, Nico; Wood, Peter

doi:10.1090/S0002-9939-00-05788-9

Diagonal type conditions on group C$^*$-algebras
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by Nico Spronk and Peter Wood PDF

Proc. Amer. Math. Soc. 129 (2001), 609-616 Request permission

Abstract:

Let $G$ be a locally compact group with $\mathbf {C}^* (G)$ and $\mathbf {C}^*_r (G)$ its enveloping and reduced C$^*$-algebras respectively. We show that if $\mathbf {C}^*(G)$ is residually finite dimensional, then $G$ is maximally almost periodic, and $\mathbf {C}^*_r (G)$ is residually finite dimensional if and only if $G$ is both amenable and maximally almost periodic. Letting $\lambda _G$ be the left regular representation of $G$, we show that a certain quasidiagonality condition on $\{\lambda _G(s):s\in G\}$ implies that $G$ is amenable.

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