Certain tracially nuclear dimensional for certain crossed product ${\rm C^*}$-algebras

Abstract

Let $\Omega$ be a class of unital ${\rm C^*}$-algebras which have the second type tracial nuclear dimensional at moat $n$ (or have tracial nuclear dimensional at most $n$). Let $A$ be an infinite dimensional unital simple ${\rm C^*}$-algebra such that $A$ is asymptotical tracially in $\Omega$. Then ${\rm T^2dim_{nuc}}(A)\leq n$ (or ${\rm Tdim_{nuc}}(A)\leq n$). As an application, let $A$ be an infinite dimensional simple separable amenable unital ${\rm C^*}$-algebra with ${\rm T^2dim_{nuc}}(A)\leq n$ (or ${\rm Tdim_{nuc}}(A)\leq n$). Suppose that $\alpha:G\to {\rm Aut}(A)$ is an action of a finite group $G$ on $A$ which has the tracial Rokhlin property. Then ${\rm T^2dim_{nuc}}({{\rm C^*}(G, A,\alpha)})\leq n$ (or ${\rm Tdim_{nuc}}$ $({{\rm C^*}(G, A,\alpha)})\leq n$).