On stable maps of operator algebras

We define a strong Morita-type equivalence ${\sim }_{\sigma \mathrm{\Delta }}$ for operator algebras. We prove that $A{\sim }_{\sigma \mathrm{\Delta }}B$ if and only if $A$ and $B$ are stably isomorphic. We also define a relation $\subset _{\sigma \Delta }$foroperatoralgebras\mathrm{.}Weprovethatif$A$and$B$are$C^*$-algebras,then$A\subset _{\sigma \Delta } B$ if and only if there exists an onto $\ast$-homomorphism $\theta :B\otimes \mathcal K \rightarrow A\otimes \mathcal K,$where$\mathcal K$ is the set of compact operators acting on an infinite dimensional separable Hilbert space. Furthermore, we prove that if $A$ and $B$ are $C^{\ast }$-algebras such that $A{\subset }_{\sigma \mathrm{\Delta }}B$ and $B{\subset }_{\sigma \mathrm{\Delta }}A$, then there exist projections $r,\widehat{r}$ in the centers of $A^{\ast \ast }$ and $B^{\ast \ast }$, respectively, such that $Ar\sim {\sigma \Delta }B\hat r$and$A (id{A^{}}-r) \sim {\sigma \Delta }B(id{B^{}}-\hat r). $