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

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

2007-04-27

arxiv.org/abs/0704.3687v2

Mathematics

Operator Algebras

Exposition slightly improved

Scientific paper

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.

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