Strong Morita equivalence of operator spaces

We introduce and examine the notions of strong Δ-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 Δ-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 Δ-equivalent operator spaces have stably isomorphic second duals and strongly Δ-equivalent TRO envelopes. In the case of unital operator spaces, strong Δ-equivalence implies stable isomorphism of the C*-envelopes.