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