An Index for Inclusions of Operator Systems

TL;DR

This paper introduces a new index invariant for inclusions of operator systems, generalizing existing invariants and demonstrating properties like multiplicativity, with computations and connections to quantum graph parameters.

Contribution

It defines a novel index for operator system inclusions, extending the quantum Lovász theta invariant and analyzing its properties and examples.

Findings

The index is multiplicative under minimal tensor products.

The invariant generalizes the quantum Lovász theta.

Examples demonstrate the invariant's computation and relevance.

Abstract

Inspired by a well-known characterization of the index of an inclusion of II$_1$ factors due to Pimsner and Popa, we define an index-type invariant for inclusions of operator systems. We compute examples of this invariant, show that it is multiplicative under minimal tensor products, and explain how it generalizes the quantum Lov\'asz theta invariant for a matricial system defined by Duan, Severini, and Winter.