Factorization of completely bounded maps through reflexive operator spaces with applications to weak almost periodicity

TL;DR

This paper establishes a factorization framework for weakly almost periodic functionals on Hopf--von Neumann algebras, demonstrating how products of such elements preserve weak almost periodicity under certain conditions.

Contribution

It introduces a new factorization result for operators through reflexive operator spaces, linking weakly compact operators and tensor products in the context of von Neumann algebras.

Findings

Product of weakly almost periodic elements remains weakly almost periodic under specific conditions.

Factorization through reflexive operator spaces is key to understanding weak almost periodicity.

Results apply to injective Hopf--von Neumann algebras, broadening the scope of weak almost periodicity analysis.

Abstract

Let $(M,\Gamma)$ be a Hopf--von Neumann algebra, so that $M_\ast$ is a completely contractive Banach algebra. We investigate whether the product of two elements of $M$ that are both weakly almost periodic functionals on $M_\ast$ is again weakly almost periodic. For that purpose, we establish the following factorization result: If $M$ and $N$ are injective von Neumann algebras, and if $x, y \in M \bar{\otimes} N$ correspond to weakly compact operators from $M_\ast$ to $N$ factoring through reflexive operator spaces $X$ and $Y$, respectively, then the operator corresponding to $xy$ factors through the Haagerup tensor product $X \otimes^h Y$ provided that $X \otimes^h Y$ is reflexive. As a consequence, for instance, for any Hopf--von Neumann algebra $(M,\Gamma)$ with $M$ injective, the product of a weakly almost periodic element of $M$ with a completely almost periodic one is again weakly almost periodic.