Co-representations of Hopf-von Neumann algebras on operator spaces other than column Hilbert space

TL;DR

This paper demonstrates that the concept of co-representations of abelian Hopf--von Neumann algebras on reflexive Banach spaces cannot be generalized beyond subhomogeneous cases, highlighting the special role of column Hilbert spaces.

Contribution

It shows the limitations of extending co-representations to general reflexive operator spaces, identifying subhomogeneity as a key condition.

Findings

Normal spatial tensor product is a Banach algebra only if the von Neumann algebra is subhomogeneous or the operator space is column Hilbert space.

The notion of co-representation cannot be extended beyond subhomogeneous Hopf--von Neumann algebras.

The result clarifies the structural constraints for co-representations on operator spaces.

Abstract

Recently, M. Daws introduced a notion of co-representation of abelian Hopf--von Neumann algebras on general reflexive Banach spaces. In this note, we show that this notion cannot be extended beyond subhomogeneous Hopf--von Neumann algebras. The key is our observation that, for a von Neumann algebra $\M$ and a reflexive operator space $E$, the normal spatial tensor product $\M \bar{\tensor} \CB(E)$ is a Banach algebra if and only if $\M$ is subhomogeneous or $E$ is completely isomorphic to column Hilbert space.