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

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.