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Recently, M. Daws introduced a notion of co-representation of abelianHopf--von Neumann algebras on general reflexive Banach spaces. In this note, weshow that this notion cannot be extended beyond subhomogeneous Hopf--vonNeumann 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 subhomogeneousor $E$ is completely isomorphic to column Hilbert space.

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