TITLE: About the category of von neumann algebras
QUESTION [23 upvotes]: I am looking for one (or more) reference about properties of the category of von Neumann algebra. 
More precisely, in an answer of a previous question, Dmitri Pavlov mentions
that the $W^*$ category is complete and cocomplete. I would be happy to have a reference for these facts. 
On a related note, at the page 84 of the volume 3 of his treatise "Theory of 
operator algebras", Takesaki says:
"Namely, 
if one insists to have the universal property for the inductive limit, then one has 
to treat a non-separable von Neumann algebra as the inductive limit of a sequence 
of separable von Neumann algebras. Therefore, we shall not consider the general 
theory of inductive limits of von Neumann algebras." 
This got me curious. Where can I find a reference for an example of an inductive system 
of separable von Neumann algebras with non separable limit ? Is it possible for this example 
to be a countable inductive system ?
And this makes me wonder : why is nonseparability such a problem in the theory of von Neumann algebra, as Takesaki suggests ?

REPLY [6 votes]: I agree with Dmitri Pavlov that separability is not so important in the modern theory of von Neumann algebras, and this answers the second question. However, an example answering the first question has not yet been given, so here is one. 
We take $\mathbf{vNA}$ to be the category of von Neumann algebras and normal *-homomorphisms, and the category $\mathbf{C}^*\mathbf{Alg}$ to be the category of unital C$^*$-algebras and *-homomorphisms. I will be using the fact that the forgetful functor $U : \mathbf{vNA} \rightarrow \mathbf{C}^*\mathbf{Alg}$ has a left adjoint $\mbox{-}^{**}$, the enveloping von Neumann algebra (the double dual of a C$^*$-algebra). 
First, observe that $C(2^\omega)$ is the colimit of the diagram
$$
C(2) \rightarrow C(2^2) \rightarrow C(2^3) \rightarrow \cdots
$$
in $\mathbf{C}^*\mathbf{Alg}$. (The easiest way I know to prove this is to define the colimiting map on the indicator functions of clopen subsets of $2^\omega$, and then use Stone-Weierstrass to extend this to all of $C(2^\omega)$.)
Each of the C*-algebras $C(2^n)$ is finite-dimensional, and therefore isomorphic to its own enveloping von Neumann algebra. Therefore the diagram above is also a diagram in $\mathbf{vNA}$, consisting of separable von Neumann algebras. Because left adjoints preserve colimits, we have that $C(2^\omega)^{**}$ is the colimit of this diagram in $\mathbf{vNA}$. 
The Dirac delta functions $\{\delta_x\}_{x \in 2^\omega}$ form an uncountable norm-discrete subset of $C(2^\omega)^*$. Therefore $C(2^\omega)^*$ is not norm-separable, so $C(2^\omega)^{**}$ does not have a separable predual, i.e. it is not a separable von Neumann algebra. But as we saw above, it is an inductive limit of finite-dimensional (and therefore separable) von Neumann algebras.