Group 𝐶*-algebras of real rank zero or one

Kaniuth, Eberhard

doi:10.1090/S0002-9939-1993-1164146-1

Group $C^ \ast$-algebras of real rank zero or one
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Proc. Amer. Math. Soc. 119 (1993), 1347-1354 Request permission

Abstract:

Let $G$ be a locally compact group and ${C^{\ast }}(G)$ its group ${C^{\ast }}$-algebra, and denote by $\operatorname {RR} ({C^{\ast }}(G))$ the real rank of ${C^{\ast }}(G)$. This note is a first step towards relating $\operatorname {RR} ({C^{\ast }}(G))$ to the structure of $G$. We identify the connected groups $G$ with $\operatorname {RR} ({C^{\ast }}(G)) = 0$ as precisely the compact connected ones and characterize the nilpotent groups whose ${C^{\ast }}$-algebras have real rank zero or one.

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