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!