Tracial approximation is stable

Abstract

Let Ω \Omega be a class of unital C ∗ ^* -algebras such that Ω \Omega is closed under tensoring with matrix algebras and taking unital hereditary C ∗ ^* -subalgebras and such that t s r ( B ) = 1 tsr(B)=1 and the Cuntz semigroup Cu ( B ) (B) is almost unperforated for any B ∈ Ω B\in \Omega . Then A ∈ A\in TA Ω \Omega for any unital C ∗ ^* -algebra A ∈ A\in TA(TA Ω ) \Omega ) . As an application, this result can be used to study tracially quasidiagonal C ∗ ^* -algebra extensions of tracial topological rank no more than one.

A Remark on Weak Tracial Approximation

“…Elliott, Fan, and Fang in [4] investigated the inheritance of several comparison properties and divisibility. Elliott and Fan [5] also proved that tracial approximation is stable for some classes of unital C * -algebras. Fu and Lin developed the notion of asymptotical tracial approximation in [6], trying to find a tracial version of Toms-Winter conjecture.…”