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

Abstract Let $\Omega $ be a class of unital $C^*$ -algebras. Then any simple unital $C^*$ -algebra $A\in \mathrm {TA}(\mathrm {TA}\Omega )$ is a $\mathrm {TA}\Omega $ algebra. Let $A\in \mathrm {TA}\Omega $ be an infinite-dimensional $\alpha $ -simple unital $C^*$ -algebra with the property SP. Suppose that $\alpha :G\to \mathrm {Aut}(A)$ is an action of a finite group $G$ on $A$ which has a certain non-simple tracial Rokhlin property. Then the crossed product algebra $C^*(G,A,\alpha )$ belongs to $\mathrm {TA}\Omega $ .