Address. Dipartimento di Matematica, Universit\'a di Catania, Viale A.\,Doria 6, 95125 Catania, ITALIA
Abstract. We show that the equality $W(E,F^\ast) = K)E, F^\ast$ is a necessary condition for the validity of certain results about isomorphic properties in the projective tensor product $E \otimes_\pi F$ of two Banach spaces under some approximation property type assumptions.
AMSclassification. 46B28, 46B03, 46B20
Keywords. Operator spaces, isomorphic properties, approximation properties