  Let ${\cal A}_1$ be the class of all unital separable simple $C^*$-algebras
$A$ such that $A\otimes U$ has tracial rank at most one for all UHF-algebras of
infinite type. It has been shown that amenable ${\cal Z}$-stable $C^*$-algebras
in ${\cal A}_1$ which satisfy the Universal Coefficient Theorem can be
classified up to isomorphism by the Elliott invariant. We show that $A\in {\cal
A}_1$ if and only if $A\otimes B$ has tracial rank at most one for one of
unital simple infinite dimensional AF-algebra $B.$ In fact, we show that $A\in
{\cal A}_1$ if and only if $A\otimes B\in {\cal A}_1$ for some unital simple
AH-algebra $B.$ Other results regarding the tensor products of $C^*$-algebras
in ${\cal A}_1$ are also obtained
