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 $C^\ast(Γ)$, such as the $K$-theoretic $K_1(C^\ast(Γ))$ 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 $C^\ast(Γ)$ (even $Γ$) is characterized by the topological group structure of its unitary group when $Γ$ is torsion-free and Abelian.

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