Symmetrisations of operator spaces

TL;DR

The paper develops a new metric tensorial construction, the balanced symmetrisation, to address Morita-type phenomena for operator systems and selfadjoint operator spaces over a unital C*-algebra. It builds the symmetrisation $\mathcal{E}^* \otimes_{\rm s}^{\mathcal{A}} \mathcal{S} \otimes_{\rm s}^{\mathcal{A}} \mathcal{E}$ via a universal property tied to $\,\mathcal{A}$-balanced c.p. trilinear maps, and proves a key factorisation (Theorem B) that yields a canonical representation $\widetilde{\theta}_{\rm s\mathcal{A}} = \phi^* \cdot \psi \cdot \phi$ (with $\psi$ unital c.p. and $\phi$ cb). The authors establish the balanced symmetrisation as a selfadjoint operator space with a natural norm $\|\cdot\|_{\rm s}$, compare it to the Haagerup norm, and develop a framework for when the symmetrisation becomes an operator system via semi-units, including concrete obstructions and examples. They further apply this machinery to function spaces, providing a crisp positivity criterion via positive semidefinite matrix-valued kernels, and derive an operator-system Morita-type factorisation theorem (Δ-equivalence) for operator systems. The results yield canonical constructions and dualities that connect operator space duals, tensorial factorisations, and Morita-type equivalences in the noncommutative setting, with implications for stable isomorphism and quantum-information-inspired notions of equivalence.

Abstract

Let $A$ be a unital C*-algebra, $S$ be an operator $A$-system and $E$ be an operator space that is a left operator $A$-module. We introduce the symmetrisation of the pair $(E,S)$ as the Hausdorff completion of the balanced tensor product $E^* \odot^{A} S \odot^{A} E$ with respect to a seminorm arising from the family of completely contractive completely positive $A$-balanced trilinear maps. We show that the symmetrisation is a selfadjoint operator space in the sense of W. Werner, possessing a universal mapping property for pairs of representations of $S$ and $E$, compatible with the $A$-module actions. We point out cases where the symmetrisation is an operator system, and where it does not admit an Archimedean order unit. We study separately the case where $A = \mathbb{C}$; in this case, we show that the symmetrisation seminorm is a norm, which is equivalent to, yet different from, the Haagerup tensor norm. When $S = \mathbb{C}$ we show that the symmetrisation is compatible with taking operator space duals. In the case where $E$ is a function space and $S = \mathbb{C}$, we characterise the positive matricial cones of the symmetrisation in terms of positive semi-definiteness of naturally associated matrix-valued functions. As an application, we provide a characterisation of Morita equivalence in the operator system category involving tensorial decomposition where the analytic structure is provided by the symmetrisation. This establishes an operator system counterpart of the factorisation Morita Theorem in other categories.

Theorems & Definitions (159)

• Theorem 2.1.1
• Theorem 2.1.2
• proof
• Remark 2.2.1
• Remark 2.2.2
• Proposition 3.1.1
• proof
• Definition 3.2.1
• Definition 3.2.2
• Theorem 3.2.3