Some properties for certain generalized tracial approximated ${\rm C^*}$-algebras

TL;DR

This paper introduces a new class of generalized tracial approximation ${ m C^*}$-algebras and demonstrates their properties, including how certain subalgebras influence the larger algebra's structure, extending previous results.

Contribution

It defines a class of generalized tracial approximation ${ m C^*}$-algebras and proves that properties like tracial $ m Z$-absorption are inherited from subalgebras.

Findings

Generalized tracial approximation ${ m C^*}$-algebras have specific structural properties.

If a subalgebra is tracially $ m Z$-absorbing, the larger algebra inherits this property.

The results extend known inheritance properties to a broader class of ${ m C^*}$-algebras.

Abstract

In this paper, we introduce a class of generalized tracial approximation ${\rm C^*}$-algebras. Let $\mathcal{P}$ be a class of unital ${\rm C^*}$-algebras which have tracially $\mathcal{Z}$-absorbing (tracial nuclear dimension at most $n$, $\rm SP$ property, $m$-almost divisible, weakly $(m, n)$-divisible). Then $A$ has tracially $\mathcal{Z}$-absorbing (tracial nuclear dimension at most $n$, $\rm SP$ property, weakly $m$-almost divisible, secondly weakly $(m, n)$-divisible) for any simple unital ${\rm C^*}$-algebra $A$ in the class of this generalized tracial approximation ${\rm C^*}$-algebras. As an application, Let $A$ be an infinite dimensional unital simple ${\rm C^*}$-algebra, and let $B$ be a centrally large subalgebra of $A$. If $B$ is tracially $\mathcal{Z}$-absorbing, then $A$ is tracially $\mathcal{Z}$-absorbing. This result was obtained by Archey, Buck and Phillips in \cite{AJN}.