Crossed products and MF algebras (Q2831913)

The authors prove that the crossed product \(\mathscr{A}\rtimes_\alpha G\) of a unital finitely generated MF algebra \(\mathscr{A}\) by a discrete finitely generated amenable residually finite group \(G\) is an MF algebra, provided that the action \(\alpha\) is almost periodic. This generalizes a result of \textit{D. Hadwin} and \textit{J.-H. Shen} [Int. J. Math. 21, No. 10, 1239--1266 (2010; Zbl 1213.46055)]. They also construct two examples of crossed product \(C^*\)-algebras whose Brown-Douglas-Fillmore extension semigroups are not groups.