On stable maps of operator algebras

Abstract

We define a strong Morita-type equivalence ${\sim }_{\sigma \Delta }$ for operator algebras. We prove that $A{\sim }_{\sigma \Delta }B$ if and only if $A$ and $B$ are stably isomorphic. We also define a relation ${\subset }_{\sigma \Delta }$ for operator algebras. We prove that if $A$ and $B$ are $C^{\ast }$-algebras, then $A{\subset }_{\sigma \Delta }B$ if and only if there exists an onto $\ast$-homomorphism $\theta :B\otimes \mathcal{K}\to A\otimes \mathcal{K},$ 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 $A{\subset }_{\sigma \Delta }B$ and $B{\subset }_{\sigma \Delta }A$, then there exist projections $r,\hat{r}$ in the centers of $A^{\ast \ast }$ and $B^{\ast \ast }$, respectively, such that $Ar{\sim }_{\sigma \Delta }B\hat{r}$ and $A(i{d}_{A^{\ast \ast }}-r){\sim }_{\sigma \Delta }B(i{d}_{B^{\ast \ast }}-\hat{r}).$