Morita equivalence for operator systems

TL;DR

This paper introduces a new equivalence relation called $\Delta$-equivalence for operator systems, showing it aligns with stable isomorphism and preserves key properties like nuclearity and representation categories.

Contribution

It defines $\Delta$-equivalence, establishes its equivalence with stable isomorphism, and explores its implications for tensor products, order isomorphisms, and graph operator systems.

Findings

$\Delta$-equivalence coincides with stable isomorphism.

Nuclearity is invariant under $\Delta$-equivalence.

Graph operator systems' $\Delta$-equivalence characterized combinatorially.

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 when represented in the double dual of their C*-envelopes.