Morita equivalence for operator systems
Abstract
We define $\Delta$-equivalence for operator systems and show that it is identical to stable isomorphism. We define $\Delta$-contexts and bihomomorphism contexts and show that two operator systems are $\Delta$-equivalent if and only if they can be placed in a $\Delta$-context, equivalently, in a bihomomorphism context. We show that nuclearity for a variety of tensor products is an invariant for $\Delta$-equivalence and that function systems are $\Delta$-equivalent precisely when they are order isomorphic. We prove that $\Delta$-equivalent operator systems have equivalent categories of representations. As an application, we characterise $\Delta$-equivalence of graph operator systems in combinatorial terms. We examine a notion of Morita embedding for operator systems, showing that mutually $\Delta$-embeddable operator systems have orthogonally complemented $\Delta$-equivalent corners.
- Publication:
-
arXiv e-prints
- Pub Date:
- September 2021
- DOI:
- 10.48550/arXiv.2109.12031
- arXiv:
- arXiv:2109.12031
- Bibcode:
- 2021arXiv210912031E
- Keywords:
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- Mathematics - Operator Algebras;
- 47L25;
- 46L07
- E-Print:
- 40 pages. Added details in the proof of Theorem 3.17 (for the implication (iv) to (i))