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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.

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    $\begingroup$ One of the distinguishing things about von Neumann algebras and C*-algebras is that they have a wealth for projections. For instance, one of the consequences is that von Neumann algebras must be unital. Later on you'll see that von Neumann algebras must be of certain types, all definitions of types being based on how the projections appear. This is the so-called "type decomposition theorem". It shows up early in von Neumann algebra theory, and thereafter one typically would study one type at a time. (The methods for each type are typically different from one another). $\endgroup$ – Munk Sep 5 '18 at 15:17
  • $\begingroup$ What leads us to the type decomposition theorem using projections? I know a von Neumann algebra is closed under the Borel calculus and that this gives a lot of projections. Are there specific examples of von Neumann algebras which von Neumann and Murray had in mind to make the distinction between different kinds of projections? $\endgroup$ – abcdef Sep 5 '18 at 15:21
  • $\begingroup$ They probably had a few in mind, albeit I am not an expert on the historical aspect. I reckon they studied the hyperfinite II-factor R rather thoroughly, B(H) of course (B(H) exhausts the type I examples), and for type III they probably had something along the lines of Araki-woods factors in mind. Actually, if I remember correctly (someone do correct me otherwise), they started their comparison theory of projections on R using the unique trace it admits. $\endgroup$ – Munk Sep 5 '18 at 15:36
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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 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.

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  • $\begingroup$ Thank you for your nice answer. May I also ask how they saw the importance of a tracial state, is it something obvious once you get deeper into the theory? Also, do you know of a reference which works more chronological historically? $\endgroup$ – abcdef Sep 6 '18 at 14:27
  • $\begingroup$ I don't have a historical answer, but the trace plays a big role with matrices, and surely matrices were an inspiration. As for books, the only book I know that (very roughly) follows the above logic is Sunder's An invitation to von Neumann Algebras (at least in the analysis of projections in a II$_1$-factor). $\endgroup$ – Martin Argerami Sep 6 '18 at 14:37

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