Crossed products of type I AF algebras by Abelian groups
Let \((G,A,\alpha)\) be a separable \(C^*\)-dynamical system, with \(G\) abelian, and let \(\Gamma\) denote the dual group of \(G\). We characterize the \(\Gamma\)-invariant ideals of the crossed product algebra \(G\times_{\alpha} A\), and use this characterization to prove that if in addition \(G\) is compact and \(A\) is type I AF, then \(G\times_{\alpha} A\) is AF also. Finally, assuming \(G\) is discrete abelian and both \(A\) and \(G\times_{\alpha} A\) are type I, we determine necessary and sufficient conditions in terms of \(A\) and the isotropy subgroups for the action of \(G\) on \(\hat A\), for \(G\times_{\alpha} A\) to be AF.
scientific article; zbMATH DE number 4054347
scientific article