TITLE: Defining the abstract tensor product of W*-algebras via a universal property
QUESTION [9 upvotes]: I am playing around a bit with $W^*$-algebras, and I'm trying to come up with a definition for the $W^*$-algebraic tensor product. Here is my first attempt:

It is easy to show that such an object exists. Simply represent the $W^*$-algebras as concrete von Neumann algebras on Hilbert spaces and consider the usual von Neumann algebra tensor product (i.e. the weak closure of the algebraic tensor product $M \odot N$).
However, I fail to show that the above definition determines the $W^*$-algebra $M \overline{\otimes} N$ uniquely (up to (normal) $*$-isomorphism). Is this true? If not, can we still modify the above universal property to get a well-working definition?
Basically, I want my universal property to convey the intuition that the $W^*$-tensor product is simply the usual von Neumann tensor product when we appropriately represent the spaces involved.
Thanks in advance for any help!

REPLY [5 votes]: Something similar was studied by Wiersma in arXiv:1506.01671 [math.OA].  However, that paper takes a different definition of the universal property: it wants the individual maps $\sigma$ and $\tau$ to be weak$^*$-homeomorphisms onto their ranges, and the map $\iota$ needs to have this property as well.
However, even here you do not recover the usual von Neumann tensor product, but rather something larger.