The tracial topological rank of $C^*$-algebras (II) Shanwen HuHuaxin LinYifeng Xue 46L05tracial rank$C^*$-algebras 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. Indiana University Mathematics Journal 2004 text pdf 10.1512/iumj.2004.53.2458 10.1512/iumj.2004.53.2458 en Indiana Univ. Math. J. 53 (2004) 1579 - 1606 state-of-the-art mathematics http://iumj.org/access/