Tensor Products of Classifiable C*-algebras

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