IUMJ

Title: The tracial topological rank of $C^*$-algebras (II)

Authors: Shanwen Hu, Huaxin Lin and Yifeng Xue

Issue: Volume 53 (2004), Issue 6, 1579-1606

Abstract:

We show that if $A$ is a unital $C^{*}$-algebra with tracial topological rank $r$ (and write $\mbox{\upshape TR}(A)=r$) and $\dim X=k$, then $\mbox{\upshape TR}(A\otimes C(X))\le r+k$. Suppose that $\mbopx{\upshape TR}(B)=k$. It is shown that $\mbox{\upshape TR}(A\otimes B)\le r+k$ if both $A$ and $B$ are assumed to be simple, or $B$ is an $AH$-algebra. Examples are also given, which show that the corresponding equality could not hold in general.