Group 𝐶*-algebras of real rank zero or one

Abstract

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