Abstract
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.
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