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 $C^*$-algebras, then $A\subset _{\sigma \Delta } B$ if and only if there exists an onto $*$-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 $C^*$-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 $A^{**}$ and $B^{**}$, respectively, such that $Ar\sim _{\sigma \Delta }B\hat r$ and $A (id_{A^{**}}-r) \sim _{\sigma \Delta }B(id_{B^{**}}-\hat r). $