Certain properties for crossed products by automorphisms with a certain non-simple tracial Rokhlin property (Q2873990)

\textit{G. A. Elliott} and \textit{Z. Niu} introduced in [J. Funct. Anal. 254, No. 2, 396--440 (2008; Zbl 1137.46035)] the class \(\roman{TA}\Omega\) of simple unital \(C^*\)-algebras which can be tracially approximated by \(C^*\)-algebras belonging to a given class \(\Omega\). The present authors extended in [Houston J. Math. 37, No. 4, 1249--1263 (2011; Zbl 1241.46033)] this notion for non-simple unital \(C^*\)-algebras. The main result of the paper under review shows that, for a given class \(\Omega\) of unital \(C^*\)-algebras, any simple unital \(C^*\)-algebra in the class \(\roman{TA}(\roman{TA}\Omega)\) is a \(\roman{TA}\Omega\) algebra. It is also proved that, for a unital \(C^*\)-algebra \(A\in\roman{TA}\Omega\) with the property SP (i.e., every non-zero hereditary \(C^*\)-subalgebra of \(A\) contains a non-zero projection) which is \(\alpha\)-simple with respect to an action \(\alpha\) of a finite group \(G\) on \(A\) which has a certain non-simple tracial Rokhlin property, the crossed product \(C^*\)-algebra \(C^*(G,A,\alpha)\) belongs to \(\roman{TA}\Omega\). Two corollaries generalize some results from \textit{X.-B. Yang} and \textit{X.-C. Fang} [Rocky Mt. J. Math. 42, No. 1, 339--352 (2012; Zbl 1253.46062)]. In the first one, \(C^*(G,A,\alpha)\) has tracial topological rank zero (\(\roman{TR}(C^*(G,A,\alpha))=0\)) provided that \(A\) is a unital \(\alpha\)-simple tracial rank-zero algebra. Secondly, if \(A\) is a unital tracial topological rank-one \(C^*\)-algebra with the property SP, then \(\roman{TR}(C^*(G,A,\alpha))\leq 1\).