Products and factorization in operator systems

TL;DR

This paper develops a framework for understanding products in unital operator spaces, providing new representation theorems, tensor product properties, and applications in quantum information theory.

Contribution

It introduces a matrix-norm characterization of partial products, studies product-respecting C*-covers, and unifies tensor products for operator systems with new factorization formulas.

Findings

Haagerup tensor product remains injective

Projectivity holds relative to product quotients

Identifies the commuting tensor product as a quotient

Abstract

We study unital operator spaces endowed with a partially defined product. We give a matrix-norm characterization of such products that allows for a representation theorem where the partial product is realized as composition of operators on Hilbert space. We study product-respecting C*-covers, including a universal product C*-cover, and product quotients. We show that for the Haagerup tensor product of unital operator spaces remains injective, while projectivity holds relative to product quotients. Moreover, we identify the commuting tensor product as a complete product quotient of the Haagerup tensor product. Our framework yields new factorization norm formulas for a variety of product structures, as well as an intrinsic trace-extension criterion that resolves a question posed by Sinclair. Our work unifies and extends tensor products for operator systems, with applications in quantum information theory.