Characterizing group $C^{\ast }$-algebras through their unitary groups: the Abelian case

We study to what extent group $C^{\ast }$-algebras are characterized by their unitary groups. A complete characterization of which Abelian group $C^{\ast }$-algebras have isomorphic unitary groups is obtained. We compare these results with other unitary-related invariants of $C^{\ast }(\mathrm{\Gamma })$, such as the $K$-theoretic $K_1(C^{\ast }(\mathrm{\Gamma }))$ and find that $C^{\ast }$-algebras of nonisomorphic torsion-free Abelian groups may have isomorphic $K_1$-groups, in sharp contrast with the well-known fact that $C^{\ast }(\mathrm{\Gamma })$ (even $\mathrm{\Gamma }$) is characterized by the topological group structure of its unitary group when $\mathrm{\Gamma }$ is torsion-free and Abelian.