Symmetrisations of operator spaces

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.