Regularity properties of certain crossed product \(C^*\)-algebras
\textit{H.-X. Lin} [Proc. Lond. Math. Soc. (3) 83, No. 1, 199--234 (2001; Zbl 1015.46031)] introduced the notion of tracial approximation by interval algebras, \textit{G. A. Elliott} and \textit{Z. Niu} [J. Funct. Anal. 254, No. 2, 396--440 (2008; Zbl 1137.46035)] generalized the notion to general \(C^*\)-algebras. Let \(\Omega\) be a class of unital \(C^*\)-algebras. Denote by TA\(\Omega\) the class of \(C^*\)-algebras which can be tracially approximated by \(C^*\)-algebras in \(\Omega\). The two notions are very useful in the classification of \(C^*\)-algebras. Moreover, $m$-comparison of positive elements property, strong tracial $m$-comparison of positive elements property and finite tracial nuclear dimension property are three important properties of \(C^*\)-algebras, and closely related to other interesting properties, such as tensorial absorption of the Jiang-Su algebra and finite nuclear dimension. In this paper, the authors expand on the list of properties that pass from a class \(\Omega\) to the class TA\(\Omega\) and prove that the above three properties of \(C^*\)-algebras in a class \(\Omega\) are inherited by simple unital \(C^*\)-algebras with SP property in the class TA\(\Omega\). Using the above results, the authors show that every unital (nuclear) simple \(C^*\)-algebras with tracial topological rank at most \(k\) has tracial nuclear dimension (nuclear dimension) at most~\(k\). Let \(A\) be an infinite dimensional separable simple unital exact \(C^*\)-algebra with the SP property and \(\alpha\) be an action of a finite group \(G\) on \(A\) with the tracial Rokhlin property. If \(A\) has one of the above three properties, so does the crossed product \(C^*\)-algebra.
scientific article; zbMATH DE number 7344138
scientific article