On stable maps of operator algebras

We define a strong Morita-type equivalence $sim _{sigma Delta }$ for operator algebras. We prove that $Asim _{sigma Delta }B$ 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 $Asubset _{sigma Delta } B$ if and only if there exists an onto $*$-homomorphism $theta :Botimes mathcal K rightarrow Aotimes mathcal K,$ 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 $Asubset _{sigma Delta } B$ and $Bsubset _{sigma Delta } A $, then there exist projections $r, hat r$ in the centers of $A^{**}$ and $B^{**}$, respectively, such that $Arsim _{sigma Delta }Bhat r$ and $A (id_{A^{**}}-r) sim _{sigma Delta }B(id_{B^{**}}-hat r). $