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Strong Morita equivalence of operator spaces. (English) Zbl 1368.46045

Summary: 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.

MSC:

46L07 Operator spaces and completely bounded maps
47L30 Abstract operator algebras on Hilbert spaces
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