AF embeddability of crossed products of AT algebras by the integers and its application
For which \(\mathbb Z^2\)-actions \(\alpha\), on a metrizable compact space \(X\), is the crossed product \(C(X) \rtimes_{\alpha} \mathbb Z^2\) embeddable into an AF-algebra? \textit{D. Voiculescu} brought up this question in [Integral Equ. Oper. Theory 17, 137-149 (1993; Zbl 1048.47501)]. Extending Brown's AF embedding theorem in a similar way [cf. \textit{N. P. Brown}, J. Funct. Anal. 160, 150-175 (1998; Zbl 0931.46043)] the author answers this question partially as follows: If \(A\) is a unital simple AT-algebra with real rank zero and \(\alpha\) is an automorphism of \(A\), then the crossed product \(C^*\)-algebra \(A\rtimes _{\alpha}\mathbb Z\) is AF embeddable. Recall that an \(AT\)-algebra is an inductive limit of finite direct sums of matrix algebras over \(C({\mathbb T})\). Another answer to the above question can be found in \textit{V. Pimsner} [Ergodic Theory Dyn. Syst. 3, 613-626 (1983; Zbl 0582.46063)]. As an application, the author proves the AF embeddability of the crossed product \(C^*\)-algebra \(C^*(X,T,S)\) arising from certain \(\mathbb Z^2\)-minimal dynamical systems \((X,T,S)\) induced by two commuting homeomorphisms \(T\) and \(S\).
scientific article; zbMATH DE number 1836668
scientific article