# Getting the gist of von Neumann algebras

I am having an issue with my studying: I am focusing on C*-algebra theory, but I am encountering many propositions that have to do with von Neumann algebras. I cannot understand the ultraweak topology, especially the fact that it is independent of the representation $$M\subset B(H)$$. I cannot understand what that means, since by the definition I have encountered von Neumann algebras are defined to live in some $$B(H)$$. I also cannot understand what we mean when we say normal representation/ functional. I would like to understand these terms but I do not know where to study from. Other notions I am encountering are the enveloping von Neumann algebra and the double dual of a $$C^*$$-algebra and the isometric isomorphism between the two.

I am looking for a reference to study these concepts but as I said, this is not my focus so I do not want to deal with all the details. A short presentation would be fine for me. Right now I have seen the definition of a von Neumann algebra as a $$*$$-subalgebra of some $$B(H)$$ that is closed in SOT, the equivalence of it being closed in WOT, the double commutant theorem, the fact that vN algebra's are always unital, the fact that they are closed under polar decomposition and that they contain many projections (i.e. they are the norm-closed linear span of their projections). I have also seen Kaplansky's density theorem. Oh, I have also seen the construction of a predual through the trace-class operators and all, but it never came in handy.

Any suggestions? I tried Kadison and Ringrose but their presentation seems way too detailed for me.

The definition of a von Neumann algebra as a subalgebra of B(H) can be confusing for beginners, since it is unclear which properties depend on H and which ones don't.

Here is a better definition due to Sakai: a von Neumann algebra is a C*-algebra that admits a predual, i.e., a Banach space whose dual space is isomorphic to the underlying Banach space of the C*-algebra.

To understand where this definition comes form, recall that the Gelfand duality establishes a contravariant equivalence of categories between commutative unital C*-algebras and compact Hausdorff topological spaces.

In complete analogy to this, there is a Gelfand duality in the measurable setting, which establishes a contravariant equivalence of categories between commutative von Neumann algebras and compact strictly localizable enhanced measurable spaces.

The latter spaces in fact include measure theory studied in a typical graduate real analysis textbook, since σ-finite spaces are strictly localizable, and Radon measures are (Marczewski) compact.

To a measurable space this duality assigns the commutative von Neumann algebra of bounded measurable functions modulo equality almost everywhere, i.e., the L$$^∞$$ space.

The predual of the von Neumann algebra is precisely the space of finite complex measures, equivalently (by the Radon–Nikodym theorem) the space of complex integrable functions modulo equality almost everywhere.

With this duality in mind, it is very easy to understand where the above condition on predual comes from because in the commutative setting, it expresses precisely the Riesz representation theorem: the dual of L$$^1$$ is L$$^∞$$.

The ultraweak topology is the weak topology induced by the predual, which is manifestly independent of any choices.

For a concise reference, I recommend Sakai's book “C*-algebras and W*-algebras”.

• Thank you for your thorough answer. You're right, defining vN alg's as subsets of B(H) yields many technicalities. On the other hand Sakai's approach is a little "out there" if you know what I mean. His book is most definitely interesting! Jun 28, 2020 at 12:07
• @JustDroppedIn: Once you're past the initial stage of setting up the theory (20–30 pages or so), there is no difference between Sakai's approach and the approach using subalgebras of B(H), since by that time you have both descriptions (and approaches) at your disposal. In Sakai's approach, a subalgebra of B(H) is nothing else than a faithful representation of a von Neumann algebra on a Hilbert space, and representations certainly are important in any approach. Jun 28, 2020 at 15:38
• To be honest, I was thinking about studying Sakai's book too, but my advisor talked me out of it in a way. I'll check it out, maybe it works out for me. Thank you very much for this conversation Jun 28, 2020 at 15:40
• @JustDroppedIn: There is nothing wrong with reading multiple books! I've read both Takesaki's 3 volumes and Sakai's book. Jun 28, 2020 at 15:41
• Wow. Studying all Takesaki's volumes sounds like an extreme task! Jun 28, 2020 at 15:42

For a relatively short (and not too heavy) introduction to these topics I recommend reading Nelson's notes, specifically the chapter on the predual, since it builds on these basic facts you know to talk about normality, and the ultra weak/strong topolgoies. https://math.berkeley.edu/~brent/files/209_notes.pdf

• Thanks, I've come across these notes before, they seem interesting. I am not accepting something as an answer yet, I would like to see more opinions. By the way, the most interesting approach I've seen is defining the predual without the introduction of trace class. It is defined with the WOT continuous functionals and a norm there, but I can't remember very well right now. Jun 28, 2020 at 12:05
• @JustDroppedIn I guess it is WOT continuous functionals on the unit ball (since that is equivalent to being a normal functional). On these type of questions there is no one answer, so there is no need to accept :) Jun 29, 2020 at 7:56