Products and factorization in operator systems

2025/11/06 by Dor-On, Adam, Russell, Travis B.
#FOS: Mathematics #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.2511.04352

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.

Citations

• Operator Systems Generated by Projections
• An abstract characterization for projections in operator systems
• Maximally entangled correlation sets
• Tensor Products of the Operator System Generated by the Cuntz Isometries
• Operator systems from discrete groups
• Nuclearity Related Properties in Operator Systems
• Quotients, exactness, and nuclearity in the operator system category
• Tensor Products of Operator Systems
• Metric characterizations of isometries and of unital operator spaces and systems
• Multipliers and dual operator algebras

Related

• Tensor Products of Operator Systems
• Tensor Products in Quantum Functional Analysis: the Non-Matricial Approach
• On maximal tensor products and quotient maps of operator systems
• Operator system quotients of matrix algebras and their tensor products
• Schur tensor product of operator spaces