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

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 $\text{M}$ and a reflexive operator space $E$, the normal spatial tensor product $\M \bar{\tensor} \CB(E)$isaBanachalgebraifandonlyif$\M$ is subhomogeneous or $E$ is completely isomorphic to column Hilbert space.