Introduction to von Neumann algebras

Question

I'm learning the basics of von Neumann algebras. Every reference on the subject I can find turns to the study of projections, introduces factors and the type classification immediately after having only barely introduced what a von Neumann algebra is.

This makes it hard to follow the references since I have no idea of what they are actually trying to do/accomplish by introducing these things.

What I'm really asking is the following:

Having introduced the definition of von Neumann algebras and the bicommutant Theorem, what are open questions one tries to solve immediately and leads to the study of projections, factors and type classification?

Examples of von Neumann algebras which have interesting properties and help to solve my question are certainly welcome.

Thanks in advance.

Answer 1

This not strictly historical, but it goes more or less like this:

• You start with $B(H)$. You think of subalgebras, so think about different closures in the natural topologies (norm, sot which is pointwise convergence, wot). You are a genius and find the Double Commutant theorem.

• Now you ask yourself about what are the possible von Neumann algebras. The obvious ones are $B(H)$, for $H$ of any dimension.

• Now, this is an exercise you can do, you prove that $B(H)\simeq B(K)$ as von Neumann algebras if and only if $\dim H=\dim K$. In the proof you'll work with matrix units, that is with projections and partial isometries.

• Next you ask yourself whether there are von Neumann algebras which are not of the form $B(H)$. You start to work with Murray and suggest this problem to him. You both know about group representations; this suggests a way to construct von Neumann algebras: start with a group $G$ and a unitary representation $\pi:G\to B(H)$, and consider $M=\pi(G)''$.

• You prove that $\pi(G)''$ has a tracial state, so it cannot be isomorphic to $B(H)$ for infinite-dimensional $H$. But if $G$ is infinite, $\pi(G)''$ is infinite-dimensional, so suddenly you have von Neumann algebras that are not isomorphic to any $B(H)$. How many are there?

• $B(H)$ is a factor, i.e. its centre is trivial. You notice that if $G$ is icc (infinite conjugacy classes), then $\pi(G)''$ is a factor.

• You study von Neumann factors which are infinite-dimensional and have a faithful tracial state (these are the II$_1$). Recall that the classification of type I factors was done using projections, and that projections are equivalent (same rank) if they have equal trace. Using an analogue of the division algorithm, you prove that that there exists projections of trace $t$ for $t\in[0,1]$. Now II$_1$-factors become interesting, they have a trace, but have no minimal projections, and you have a notion of "continuous dimension".

• You ask yourself if all II$_1$-factors are isomorphic. Using knowledge about free groups (like the kind used in the Banach-Tarski paradox), you prove that $\pi(S_\infty)''$ and $\pi(\mathbb F_2)''$ are not isomorphic. Now you have the interesting question how many II$_1$-factors are there.

• You define tensor products, and now you have $\pi(S_\infty)\otimes B(H)$, which behaves like a mixture of types II$_1$ and I. These are the II$_\infty$.

• By now projections have played a big role, so you think more about comparison of projections, and you classify von Neumann factors in types I,II, III.

• You construct crossed products and you realize examples of factors of all types I, II$_1$, II$_\infty$, III , of the form $L^\infty(X)\rtimes G$ for a measure space $X$ and a group $G$ of automorphisms of $X$.

• You develop a notion of direct integral to show that any von Neumann algebra is a direct integral of factors.

This was (very, very, very) roughly the situation by the early 1940s.