The predual of W∗-tensor products over W∗-subalgebras (separable case)

Let M, N, R be W∗-algebras, with R unitally embedded in both M and N. By using Reduction Theory, we describe the predual (M⊗̄RN)∗ of the W∗-tensor product M⊗̄RN over the common W∗-subalgebra R in the separable predual case. We also analyze the case when R is a direct sum of full matrix algebras, without any separability assumption. In the last situation, the predual (M⊗̄RN)∗ is described by the center Z(RM∗⊗N∗R) of the R–R bimodule RM∗⊗N∗R, the last one being isomorphic to the predual of M⊗̄N. It is also shown that such a reduction-free result cannot be extended to the remaining cases.