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Characterizing group \(C^*\)-algebras through their unitary groups: the abelian case. (English) Zbl 1200.19003

The paper is devoted to the characterization of some group \(C^*\)-algebras of discrete abelian groups by its unitary group \(\mathcal U(C^*(\Gamma))\). The main result (Theorem 4.8) states that two countable discrete abelian groups \(\Gamma_1\) and \(\Gamma_2\) have topologically isomorphic unitary groups \(\mathcal U(C^*(\Gamma_1)) \cong \mathcal U(C^*(\Gamma_2))\) if and only if their torsion groups are isomorphic, \([t\Gamma_1]] = [t\Gamma_2] = \alpha\) and the quotient groups modulo torsions are isomorphic
\[ \bigoplus_\alpha \Gamma_1/t\Gamma_1 \cong \bigoplus_\alpha \Gamma_2/t\Gamma_2. \]
This invariant \(\mathcal U(C^*(\Gamma))/\mathcal U_0(C^*(\Gamma))\) is therefore stronger than the \(K_1\)-invariant \(K_1(C^*(\Gamma))\).

MSC:

19L99 Topological \(K\)-theory
43A40 Character groups and dual objects
46L05 General theory of \(C^*\)-algebras
46L80 \(K\)-theory and operator algebras (including cyclic theory)
19B99 Whitehead groups and \(K_1\)
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