Strong Morita equivalence of operator spaces

TL;DR

The paper introduces strong $ riangle$-equivalence and strong TRO equivalence for operator spaces, establishing their properties and relations to stable isomorphism and duals, paralleling Morita theory for C*-algebras.

Contribution

It defines and analyzes strong $ riangle$-equivalence and strong TRO equivalence for operator spaces, extending Morita theory concepts to this setting.

Findings

Strong $ riangle$-equivalence coincides with stable isomorphism under countability.

Strongly TRO equivalent spaces have isomorphic second duals.

Strong $ riangle$-equivalence implies stable isomorphism of C*-envelopes for unital spaces.

Abstract

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.