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 ?
Best Answer
The standard reference for such matters is Guichardet's paper Sur la catégorie des algèbres de von Neumann. Bulletin des Sciences mathématiques 90 (1966), 41–64. PDF file: http://math.berkeley.edu/~pavlov/scans/guichardet.pdf
I don't think separability (or the more general property of σ-finiteness) is important. Requiring it is more of a tradition than a real necessity, similar to requiring smooth manifolds to be second countable instead of just being paracompact