  We introduce and examine the notions of strong $\Delta$-equivalence and
strong TRO equivalence for operator spaces. We show that they behave in an
analogous way to how strong Morita equivalence does for the category of
C*-algebras. In particular, we prove that strong $\Delta$-equivalence coincides
with stable isomorphism under the expected countability hypothesis, and that
strongly TRO equivalent operator spaces admit a correspondence between
particular representations. Furthermore we show that strongly
$\Delta$-equivalent operator spaces have stably isomorphic second duals and
strongly $\Delta$-equivalent TRO envelopes. In the case of unital operator
spaces, strong $\Delta$-equivalence implies stable isomorphism of the
C*-envelopes.

Operator Algebras