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

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