Take \(\mathfrak{M, N, R}\) to be W\(^*\)-algebras with \(\mathfrak{R}\) unitally embedded in both \(\mathfrak{M}\) and \(\mathfrak{N}\). These are represented as (concrete) von Neumann algebras \(M\), \(N\) on Hilbert spaces \(H_{1}, H_{2}\) respectively, and \(R\) will correspond to a representation \(\pi\) of \(\mathfrak{R}\) in a Hilbert space \(H\). S. Stratila and L. Zsido [J. Funct. Anal. 165, No. 2, 293–346 (1999; Zbl 0951.46031)] defined the tensor product \(\mathfrak{M} \overline{\otimes}_{\mathfrak{R}}\mathfrak{N}\) giving conditions for it to exist and to be unique. Amongst these is the requirement that the tensor product is generated by \(M\) and \(N\) and that \(\pi(M) \subset \pi (R) \wedge M_{1}\) and \(\pi(N) \subset \pi(N) \wedge N_{1}\), where \(M_{1}, N_{1} \subset \pi(R)'\) are commuting type I von Neumann algebras whose common centre coincides with the centre of \(\pi(R)\). The tensor product reduces to the spatial W*-tensor product \(M \overline{\otimes} N\) when \(R = \mathbb C\).
The (Banach space) predual \((M \overline{\otimes} N)_*\) was identified with the projective tensor product \(M_{*} \widehat{\otimes} N_{*}\) by E. Effros and Z.-J. Ruan [Int. J. Math. 1, No. 2, 163–187 (1990; Zbl 0747.46014)]. The author’s aim is similarly to identify \((M \overline{\otimes}_{R} N)_{*}\). A basic tool throughout is the reduction of von Neumann algebras into components; for example \(M_1\) and \(M_2\) can be decomposed into homogeneous parts and \(\mathfrak{M} \overline{\otimes}_{\mathfrak{R}}\mathfrak{N}\) is decomposed into components of the form \(\widetilde{M}\overline{\otimes}R \overline{\otimes} \widetilde{N}\), where \(\widetilde{M}, \widetilde{N}\) are von Neumann algebras on \(H_{1}, H_{2}\) such that \(M = \widetilde{M} \overline{\otimes} R \) and \(N = R \overline{\otimes}\widetilde{N}\), respectively. In an algebraic situation, \(\widetilde{M} = M\) and \(\widetilde{N} = N\).
In the first part, the author deals with the structure of the predual of the W*-tensor product for von Neumann algebras, assuming that both \(M\) and \(N\), or \(\mathfrak{M} \overline{\otimes}_{\mathfrak{R}}\mathfrak{N}\), have separable preduals. This is a bit confusing as all that is needed are separable Hilbert spaces for the von Neumann algebras in order to be able to disintegrate them over some standard Borel measure space. A W*-algebra is countably generated if its predual is separable; the von Neumann algebra thus has a faithful normal state so that, though it may be on an inseparable Hilbert space, one can construct an isomorphic von Neumann algebra on a separable Hilbert space.
There is a standard Borel measure space \((\Gamma, \mu)\) and a common direct integral disintegration of \(M, N, R\) with \(R\) being disintegrated into a measurable field of factors. Then \(\mathfrak{M} \overline{\otimes}_{\mathfrak{R}}\mathfrak{N}\) can be disintegrated as a direct integral of measurable fields \(\widetilde{M}(\gamma)\overline{\otimes}R(\gamma) \overline{\otimes} \widetilde{N}(\gamma), \gamma \in \Gamma\). Interpreting the latter integrand as fields of linear functionals, the predual will be a weakly-defined direct integral of the fields \(\widetilde{M}(\gamma)_{*} \widehat{\otimes} R(\gamma)_{*} \widehat{\otimes} \widetilde{N}(\gamma)_{*}\) [see M. Takesaki, Theory of operator algebras. I, Berlin: Springer-Verlag (1979; Zbl 0436.46043)].
The author deals with a non-separable case where \(R\) has atomic centre (and so is a direct sum of full matrix algebras) so that, using reductions, the proof reduces mainly to standard algebraic techniques for the tensor product of modules \(M,N\) over a commutative ring \(R\) and a (balanced) bilinear mapping \( M \times N \mapsto R\). After reducing \(R\) to a component \(\mathbf{M}_{d}\) one can reduce \(M \overline{\otimes}_{\mathbf{M}_{d}} N\) to sums of terms of the form \(\mathbf{M}_{j} \otimes \mathbf{M}_{d} \otimes \mathbf{M}_{k}\) to effect the proof; this decomposition was not clearly presented in the article. The author identifies the annihilator and the polar of \(S \subset M \overline{\otimes} N\), the *-weakly closed span of the (balanced) elements \(\{ ar \otimes b - a \otimes br \}\), as it forms a cone. He then uses the bipolar theorem to show that \({M \overline{\otimes}_{R} N} = {(M \overline{\otimes} N)/ S}\). He shows also that \((M \overline{\otimes}_{R} N)_*\) can be identified with the centre of \(M_{*} \widehat{\otimes} N_{*}\).
The author is economical with proofs and explanations; this saves paper but entails a lot of work on the part of the reader. In his outlook section 4, the author shows that this procedure of quotienting out a suitable subspace of the spatial tensor product will not work in a more general setting; he suggests the use of non-commutative L\(^p\)-spaces.