According to definition 1.22.10 in Sakai's book if $\mathscr{N}$ and $\mathscr{M}$ are two W$^*$-algebras its tensor product is defined by \begin{equation} \mathscr{N} \overline{\otimes}\mathscr{M} := (\mathscr{N}_* \otimes_{min^*} \mathscr{M}_*)^*, \end{equation} where $\text{min}^*$ is the dual projective norm and $\mathscr{N}_*$, $\mathscr{M}_*$ are the corresponding predual spaces of $\mathscr{N}$ and $\mathscr{M}$. What I would like to know if it is true that $\mathscr{N} \overline{\otimes}\mathscr{M} \cong \mathscr{M} \overline{\otimes}\mathscr{N}$ for this particular norm, that I will call W$^*$-norm, .
According to proposition II.9.2.6 in Blackadar's book when $\mathscr{N}$ and $\mathscr{M}$ are C$^*$-algebras it is true that \begin{equation} \mathscr{N} {\otimes}_{min \\ max} \mathscr{M} \cong \mathscr{M} {\otimes}_{min \\ max} \mathscr{N}, \end{equation}
however, he does not prove this result and a similar property for the case of W$^*$-algebras is not discussed.
In order to prove this isomorphism of W$^*$-algebras I started by defining the map
\begin{equation} s: \mathscr{N} \odot \mathscr{M} \to \mathscr{M} \odot \mathscr{N}, \sum_i n_i \odot m_i \mapsto \sum_i m_i \odot n_i, \end{equation}
where $\odot$ denotes the algebraic tensor product; this map has an inverse given by \begin{equation} s': \mathscr{M} \odot \mathscr{N} \to \mathscr{N} \odot \mathscr{M}, \sum_i m_i \odot n_i \mapsto \sum_i n_i \odot m_i. \end{equation} Now, the idea will be to prove that the map $s$ can be extended to an unique norm-preserving map in $\mathcal{B}(\mathscr{N} \overline{\otimes}\mathscr{M}, \mathscr{M} \overline{\otimes}\mathscr{N})$; for that I should prove that $s: \mathscr{N} \otimes_\alpha \mathscr{M} \to \mathscr{M} \overline{\otimes} \mathscr{N}$ is bounded where, $\otimes_\alpha$ is the W$^*$-norm. Then, if the same is done for the map $s': \mathscr{M} \otimes_\alpha \mathscr{N} \to \mathscr{N} \overline{\otimes} \mathscr{M}$ then, I am almost done. However, I am failing to prove that the map $s$ is bounded. One thing I do know is that since $\alpha$ is a cross norm then $||s(m\otimes n)|| = ||m \otimes n||$ but this only works for simple tensors.
So, I was wondering if there is any way I can bound these maps so they can be extended? Or perhaps, if there is another way to proceed to prove the isomorphism by using Sakai's language.
Thanks in advance.
First let me note that the treatment of tensor products in Sakai's book (and also Takesaki's book) is a bit outdated by now. These have been far better understood by now in terms of so-called operator spaces. If you really want to define the tensor product of von Neumann algebras abstractly, then take the result of Effros and Ruan that says $M\overline{\otimes}N\cong (M_\ast\hat\otimes N_\ast)^\ast$, where $\hat\otimes$ denotes the projective operator space tensor product. Here the symmetry is clear (if you know the definition of the projective operator space tensor product, that is).
That being said, think it's far more convenient to work with Prop. 1.22.11 in Sakai's book, which says that $M\overline{\otimes}N$ is isomorphic to the weak$^\ast$ closure of $\pi(M)\odot\sigma(N)$ inside $B(H\otimes K)$ whenever $\pi\colon M\to B(H)$ and $\sigma\colon N\to B(K)$ are faithful normal representations. The twist $U\colon H\otimes K\to K\otimes H$ that maps $\xi\otimes\eta$ to $\eta\otimes\xi$ is unitary, and $U(\pi(M)\odot \sigma(N))U^\ast=\sigma(N)\odot \pi(M)$. Thus conjugation with $U$ extends to a normal $\ast$-isomorphism between the weak$^\ast$ closures of $\pi(M)\odot \sigma(N)$ and $\sigma(N)\odot \pi(M)$. Therefore $M\overline{\otimes} N$ and $N\overline{\otimes}N$ are isomorphic.