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 (Mcircle times(R)N), of the W*-tensor product Mcircle times(R)N 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 (Mcircle times(R)N)(*) is described by the center Z(M-R circle times N (*) (R)) of the R-R bimodule M-R(*) circle times N-* (R). the last one being isomorphic to the predual of M circle times N. It is also shown that such a reduction-free result cannot be extended to the remaining cases. (C) 2003 Elsevier Inc. All rights reserved.