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

Abstract

Let $\Omega$ be a class of unital ${\mathrm{C}}^{\ast }$-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 ${\mathrm{C}}^{\ast }$-algebra such that $A$ is asymptotical tracially in $\Omega$. Then ${\mathrm{T}}^2\mathrm{di}{\mathrm{m}}_{\mathrm{nuc}}(A)\le n$ (or $\mathrm{Tdi}{\mathrm{m}}_{\mathrm{nuc}}(A)\le n$). As an application, let $A$ be an infinite dimensional simple separable amenable unital ${\mathrm{C}}^{\ast }$-algebra with ${\mathrm{T}}^2\mathrm{di}{\mathrm{m}}_{\mathrm{nuc}}(A)\le n$ (or $\mathrm{Tdi}{\mathrm{m}}_{\mathrm{nuc}}(A)\le n$). Suppose that $\alpha :G\to \mathrm{Aut}(A)$ is an action of a finite group $G$ on $A$ which has the tracial Rokhlin property. Then ${\mathrm{T}}^2\mathrm{di}{\mathrm{m}}_{\mathrm{nuc}}({\mathrm{C}}^{\ast }(G,A,\alpha ))\le n$ (or $\mathrm{Tdi}{\mathrm{m}}_{\mathrm{nuc}}$ $({\mathrm{C}}^{\ast }(G,A,\alpha ))\le n$).

arXiv commentarXiv admin note: text overlap with arXiv:2101.11921