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

In this paper, we introduce a class of generalized tracial approximation Calgebras. Let P be a class of unital C-algebras which have tracially Z-absorbing (tracial nuclear dimension at most n, SP property, m-almost divisible, weakly (m,n)-divisible). Then A has tracially Z-absorbing (tracial nuclear dimension at most n, SP property, weakly m-almost divisible, secondly weakly (m,n)-divisible) for any simple unital Calgebra A in the class of this generalized tracial approximation C-algebras. As an application, Let A be an infinite dimensional unital simple C-algebra, and let B be a centrally large subalgebra of A. If B is tracially Z-absorbing, then A is tracially Z-absorbing. This result was obtained by Archey, Buck and Phillips in [2].

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