Some properties for certain generalized tracial approximated ${\mathrm{C}}^{\ast }$-algebras

Abstract

In this paper, we introduce a class of generalized tracial approximation ${\mathrm{C}}^{\ast }$-algebras. Let $\mathcal{P}$ be a class of unital ${\mathrm{C}}^{\ast }$-algebras which have tracially $\mathcal{Z}$-absorbing (tracial nuclear dimension at most $n$, $\mathrm{SP}$ property, $m$-almost divisible, weakly $(m,n)$-divisible). Then $A$ has tracially $\mathcal{Z}$-absorbing (tracial nuclear dimension at most $n$, $\mathrm{SP}$ property, weakly $m$-almost divisible, secondly weakly $(m,n)$-divisible) for any simple unital ${\mathrm{C}}^{\ast }$-algebra $A$ in the class of this generalized tracial approximation ${\mathrm{C}}^{\ast }$-algebras. As an application, Let $A$ be an infinite dimensional unital simple ${\mathrm{C}}^{\ast }$-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}.

There were some errors and we have major revised

References in corpus (2)

• Nuclear dimension of simple C*-algebras
• The C*-algebra of a minimal homeomorphism with finite mean dimension has finite radius of comparison