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

TL;DR

This paper investigates the preservation of tracial nuclear dimension in certain crossed product ${ m C^*}$-algebras, showing that under specific conditions, the dimension remains bounded after group actions with the tracial Rokhlin property.

Contribution

It establishes that tracial nuclear dimension is preserved under asymptotic tracial approximation and finite group actions with the tracial Rokhlin property.

Findings

Tracial nuclear dimension remains bounded under asymptotic tracial approximation.

Finite group actions with the tracial Rokhlin property preserve the dimension bound.

Provides bounds for the nuclear dimension of crossed product ${ m C^*}$-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 ${\rm Tdim_{nuc}}$ $({{\rm C^*}(G, A,\alpha)})\leq n$).