Generalized Tracially Approximated C*-algebras

TL;DR

This paper introduces generalized classes of tracially approximated C*-algebras and demonstrates that properties like tracial $ m Z$-absorption are preserved under certain conditions, extending previous results in the field.

Contribution

The paper defines new classes of generalized tracial approximation C*-algebras and proves the inheritance of properties like tracial $ m Z$-absorption from subalgebras to larger algebras.

Findings

Tracial properties are preserved in generalized tracial approximation classes.

If a subalgebra has a property, the larger algebra inherits it under certain conditions.

Extends previous results on property inheritance in simple unital C*-algebras.

Abstract

In this paper, we introduce some classes of generalized tracial approximation ${\rm C^*}$-algebras. Consider the class of unital ${\rm C^*}$-algebras which are tracially $\mathcal{Z}$-absorbing (or have tracial nuclear dimension at most $n$, or have the property $\rm SP$, or are $m$-almost divisible). Then $A$ is tracially $\mathcal{Z}$-absorbing (respectively, has tracial nuclear dimension at most $n$, has the property $\rm SP$, is weakly ($n, m$)-almost divisible) for any simple unital ${\rm C^*}$-algebra $A$ in the corresponding class of 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}.