Characterizing group C*-algebras through their unitary groups: the abelian case, preprint

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 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. 1