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Products and factorization in operator systems

Adam Dor-On, Travis B. Russell

TL;DR

This work develops a unified framework for unital operator spaces endowed with a partial multiplication, encoding multiplicative data intrinsically via matrix norms and factorization norms. It establishes a representation theorem that realizes the partial product as operator composition on a Hilbert space, and introduces universal product C*-covers and product quotients to respect this structure. The Haagerup tensor product is shown to be injective on unital spaces and projective with respect to product quotients, while the commuting tensor product is characterized as a complete product quotient of Haagerup; a robust factorization-norm machinery yields intrinsic norms for a variety of product structures, including operator systems, groups, and projections, and leads to a trace-extension criterion resolving Sinclair. The framework yields new descriptions of synchronous quantum commuting correlations and provides operator-space formulations of these and related tensor-product constructions with potential applications in quantum information theory and operator algebra. Together these results advance a coherent, multiplicative perspective on operator-system tensor products and their connections to quantum correlations and information processing.

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.

Products and factorization in operator systems

TL;DR

This work develops a unified framework for unital operator spaces endowed with a partial multiplication, encoding multiplicative data intrinsically via matrix norms and factorization norms. It establishes a representation theorem that realizes the partial product as operator composition on a Hilbert space, and introduces universal product C*-covers and product quotients to respect this structure. The Haagerup tensor product is shown to be injective on unital spaces and projective with respect to product quotients, while the commuting tensor product is characterized as a complete product quotient of Haagerup; a robust factorization-norm machinery yields intrinsic norms for a variety of product structures, including operator systems, groups, and projections, and leads to a trace-extension criterion resolving Sinclair. The framework yields new descriptions of synchronous quantum commuting correlations and provides operator-space formulations of these and related tensor-product constructions with potential applications in quantum information theory and operator algebra. Together these results advance a coherent, multiplicative perspective on operator-system tensor products and their connections to quantum correlations and information processing.

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.

Paper Structure

This paper contains 14 sections, 43 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Haagerup tensor product for unital operator spaces
  4. Products of unital operator spaces
  5. Unital operator spaces
  6. General partial products
  7. Universal product C*-cover and product quotients
  8. Factorization norms and their applications
  9. Unital operator spaces
  10. Commuting tensor product
  11. Group operator systems
  12. Operator systems spanned by projections
  13. Tracial states
  14. Corners of synchronous correlations

Key Result

Theorem 2.1

Let ${\mathcal{V}}$ be an abstract operator space (respectively, an operator system) with matrix norm $\{\|\cdot\|_{n,m}\}$. Then there exists a Hilbert space ${\mathcal{H}}$ and a complete isometry (respectively, a unital complete isometry) $\varphi: {\mathcal{V}} \to {\mathbb{B}}({\mathcal{H}})$.

Theorems & Definitions (86)

  • Theorem 2.1: Ruan, Choi-Effros
  • Theorem 2.2
  • Theorem 2.3
  • Corollary 2.4
  • proof
  • Proposition 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • ...and 76 more