Weakly almost periodic functionals, representations, and operator spaces

TL;DR

This paper extends the concept of weakly almost periodic functionals to operator spaces, providing an intrinsic characterization of when a completely contractive Banach algebra can be embedded into operators on a reflexive operator space.

Contribution

It adapts Xu's interpolation ideas to characterize when a Banach algebra is representable on a reflexive operator space using weakly almost periodic functionals.

Findings

Characterization of Banach algebras as subalgebras of operators on reflexive spaces

Extension of Xu's interpolation approach to operator spaces

Analogue of Young's result for operator algebras

Abstract

A theorem of Davis, Figiel, Johnson and Pe{\l}czy\'nski tells us that weakly-compact operators between Banach spaces factor through reflexive Banach spaces. The machinery underlying this result is that of the real interpolation method, which has been adapted to the category of operator spaces by Xu, showing the this factorisation result also holds for completely bounded weakly-compact maps. In this note, we show that Xu's ideas can be adapted to give an intrinsic characterisation of when a completely contractive Banach algebra arises as a closed subalgebra of the algebra of completely bounded operators on a reflexive operator space. This result was shown by Young for Banach algebras, and our characterisation is a direct analogue of Young's, involving weakly almost periodic functionals.