Super Operator Systems, Strong Norms, and Operator Tensor Products

TL;DR

This paper introduces super operator systems, extending traditional operator systems to include graded spaces with superinvolutions, and explores their tensor product relations with C*-algebras.

Contribution

It defines super operator systems and applies this framework to analyze tensor products in operator space and C*-algebra theory.

Findings

Super operator systems generalize classical operator systems.

Representation via bounded operators on Z_2-graded Hilbert spaces is established.

Relations between various tensor products and super operator systems are investigated.

Abstract

A notion of super operator system is defined which generalizes the usual notion of operator systems to include certain unital involutive operator spaces which cannot be represented completely isometric as a concrete operator system on some Hilbert space. They can nevertheless be represented by bounded operators on a standard Z_2-graded Hilbert space equipped with a superinvolution. We apply this theory to investigate on the relation between certain tensor products defined for operator spaces and C^*-algebras, such as the projective tensor product, the Haagerup tensor product and the maximal C^*-tensor product.