WHEN CAN THE STABLE ALGEBRA DETERMINE THE STRUCTURE OF A C^＊-ALGEBRA ？

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WHEN CAN THE STABLE ALGEBRA DETERMINE THE STRUCTURE OF A C*-ALGEBRA

Let $A$ and $B$ be $C^*$-algebras. Suppose that ${\Cal K}$ is the algebra of all compact operators on a seperable Hilbert space, and $\alpha$ is an action on the stable algebra ${\Cal K}\otimes A$ induced by $SU(\infty)$. It is proved that if $A$ is $\alpha$-invariant stable isomorphic to $B$, then there is a *-isomorphism between $A$ and $B$. An analogous result is obtained by considering $O_n\otimes{\Cal K}\otimes A$ in the place of ${\Cal K} \otimes A$, where $O_n$ is the Cuntz algebra $(3\le n <\infty)$.