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Morita equivalence for operator systems

George K. Eleftherakis, Evgenios T. A. Kakariadis, Ivan G. Todorov

Abstract

We define $Δ$-equivalence for operator systems and show that it is identical to stable isomorphism. We define $Δ$-contexts and bihomomorphism contexts and show that two operator systems are $Δ$-equivalent if and only if they can be placed in a $Δ$-context, equivalently, in a bihomomorphism context. We show that nuclearity for a variety of tensor products is an invariant for $Δ$-equivalence and that function systems are $Δ$-equivalent precisely when they are order isomorphic. We prove that $Δ$-equivalent operator systems have equivalent categories of representations. As an application, we characterise $Δ$-equivalence of graph operator systems in combinatorial terms. We examine a notion of Morita embedding for operator systems, showing that mutually $Δ$-embeddable operator systems have orthogonally complemented $Δ$-equivalent corners when represented in the double dual of their C*-envelopes.

Morita equivalence for operator systems

Abstract

We define -equivalence for operator systems and show that it is identical to stable isomorphism. We define -contexts and bihomomorphism contexts and show that two operator systems are -equivalent if and only if they can be placed in a -context, equivalently, in a bihomomorphism context. We show that nuclearity for a variety of tensor products is an invariant for -equivalence and that function systems are -equivalent precisely when they are order isomorphic. We prove that -equivalent operator systems have equivalent categories of representations. As an application, we characterise -equivalence of graph operator systems in combinatorial terms. We examine a notion of Morita embedding for operator systems, showing that mutually -embeddable operator systems have orthogonally complemented -equivalent corners when represented in the double dual of their C*-envelopes.

Paper Structure

This paper contains 18 sections, 27 theorems, 290 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Operator spaces
  4. Ternary rings of operators
  5. TRO envelopes
  6. Multipliers of operator spaces
  7. Morita equivalence for operator spaces
  8. $\Delta$-equivalence and its characterisations
  9. TRO-equivalence
  10. $\Delta$-equivalence
  11. Morita contexts for operator systems
  12. Categorical implications
  13. Morita equivalence invariants
  14. Isomorphism of lattices of kernels
  15. Tensor products
  16. ...and 3 more sections

Key Result

Proposition 3.3

The concrete operator systems ${\mathcal{S}}\subseteq \mathcal{B}(H)$ and ${\mathcal{T}} \subseteq \mathcal{B}(K)$ are TRO-equivalent if and only if there exists a TRO ${\mathcal{M}} \subseteq {\mathcal{B}}(H, K)$ such that $[{\mathcal{M}}^* {\mathcal{T}} {\mathcal{M}}] = \overline{{\mathcal{S}}}$ a

Theorems & Definitions (73)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 3.1
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • ...and 63 more