We study whether the operator space $V**\stackrel{\alpha }{\otimes }W**$ can be identified with a subspace of the bidual space $\left(V\stackrel{\alpha }{\otimes }W\right)**$, for a given operator space tensor norm. We prove that this can be done if α is finitely generated and V and W are locally reflexive. If in addition the dual spaces are locally reflexive and the bidual spaces have the completely bounded approximation property, then the identification is through a complete isomorphism. When α is the projective, Haagerup or injective norm, the hypotheses can be weakened.