CO-REPRESENTATIONS OF HOPF-VON NEUMANN ALGEBRAS ON OPERATOR SPACES OTHER THAN COLUMN HILBERT SPACE

Recently, 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 (circle times) over bar CB(E) is a Banach algebra if and only if m is subhomogeneous or E is completely isomorphic to column Hilbert space.