Factorization of completely bounded maps through reflexive operator spaces with applications to weak almost periodicity

Let ( M , Γ ) be a Hopf–von Neumann algebra, so that M ⁎ is a completely contractive Banach algebra. We investigate whether the product of two elements of M that are both weakly almost periodic functionals on M ⁎ is again weakly almost periodic. For that purpose, we establish the following factorization result: If M and N are injective von Neumann algebras, and if x , y ∈ M ⊗ ¯ N correspond to weakly compact operators from M ⁎ to N factoring through reflexive operator spaces X and Y, respectively, then the operator corresponding to xy factors through the Haagerup tensor product X ⊗ h Y provided that X ⊗ h Y is reflexive. As a consequence, for instance, for any Hopf–von Neumann algebra ( M , Γ ) with M injective, the product of a weakly almost periodic element of M with a completely almost periodic one is again weakly almost periodic.