Characterizing group C*-algebras through their unitary groups: the Abelian case

Abstract

We study to what extent group C*-algebras are characterized by their unitary groups. A complete characterization of which Abelian group C*-algebras have isomorphic unitary groups is obtained. We compare these results with other unitary-related invariants of C*(Γ), such as the K-theoretic K1(C*(Γ)) and find that C*-algebras of nonisomorphic torsion-free Abelian groups may have isomorphic K1-groups, in sharp contrast with the well-known fact that C*(Γ) (even Γ) is characterized by the topological group structure of its unitary group when Γ is torsion-free and Abelian