  We prove that two dual operator spaces $X$ and $Y$ are stably isomorphic if
and only if there exist completely isometric normal representations $\phi$ and
$\psi$ of $X$ and $Y$, respectively, and ternary rings of operators $M_1, M_2$
such that $\phi (X)= [M_2^*\psi (Y)M_1]^{-w^*}$ and $\psi (Y)=[M_2\phi
(X)M_1^*].$ We prove that this is equivalent to certain canonical dual operator
algebras associated with the operator spaces being stably isomorphic. We apply
these operator space results to prove that certain dual operator algebras are
stably isomorphic if and only if they are isomorphic. We provide examples
motivated by CSL algebra theory.
