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Are there simple, perhaps concrete and/or constructive examples of type III von Neumann algebra factors? By simple I mean a subset of a matrix space or of the operators of a function space. We know the hyperfinite II$_1$ factor can be represented as matrices in a very concrete and didactic way. I wonder if the type III factors could be presented as easily. If not, what are the easiest ways to illustrate the type III factors?

The main concern for me is to be able to picture the projections in those algebras, hopefully in an explicit way.

Further, could those examples be eventually extended to illustrate all $\lambda$ classes of III$_{\lambda}$ factors?

Thanks in advance for all the answers or for references pointing me to the right track.

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  • $\begingroup$ Some googling gives this: en.wikipedia.org/wiki/Tomita%E2%80%93Takesaki_theory $\endgroup$ Oct 11, 2020 at 4:48
  • $\begingroup$ Thanks for the comment and for the reference, Qiaochu. It’s a time I’ve been around von Neumann algebras. I realized from the beginning that it is a very serious and deep subject. I’m still trying to get used to the theory. I sincerely don’t know if simple examples of type III factor are available or not. I’m going to edit the question in order to better define what I mean by simple. $\endgroup$
    – Lambda
    Oct 12, 2020 at 19:22
  • $\begingroup$ I’ve made some progress, but I think tomita-Takesaki theory is beyond what I could handle right now. I sincerely don’t know if simple examples of type III factors are available or not. With an example in mind, I can better navigate through the theory and improve my intuition. Thanks. $\endgroup$
    – Lambda
    Oct 12, 2020 at 19:31

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Having spent many hours of my life thinking and learning about II$_1$ factors, I strongly think that the premise of the question is misguided. You say that the hyperfinite II$_1$ factor "can be represented as matrices in a very concrete and didactic way". While that's definitely not true, I assume that what you mean is that the hyperfinite II$_1$-factor can be seen as the sot-closure of UHF$(2^\infty)$ (that is, the C$^*$-algebra generated by the unital inclusions $M_{2^n}(\mathbb C)\subset M_{2^{n+1}}(\mathbb C) )$. The sot-closure in this situation is taken in the GNS representation of the trace.

The reason I say that above is "misguided", is that now you could take the same "matricial" C$^*$-algebra UHF$(2^\infty)$ but now consider, instead of the trace, the state induced by the weighted traces $$ \psi(A)=\sum_{j=1}^{2^n}\frac{\alpha_j A_{jj}}{(1+\lambda)^n}\qquad A\in M_{2^n}(\mathbb C) $$ where $\lambda\in(0,1)$ is fixed and the $\alpha_j$ are $1,\lambda,\ldots,\lambda^n$ in a certain order and with adequate repetitions (so that they give you precisely the terms in the expansion of $(1+\lambda)^n$; this is usually defined in a different way, but it is easy to see who the $\alpha_j$ need to be). If you do GNS for this state, on the same "matricial" C$^*$-algebra UHF$(2^\infty)$ as before, now you get a type $\text{III}_\lambda$ AFD factor. These are the Powers' Factors.

I'm not entirely sure what you mean by "subset of a matrix space" but type II and III von Neumann algebras cannot be finite-dimensional, as they don't have minimal projections.

You seem to looking for "explicit" presentations of von Neumann algebras. Not going to happen. Even in the hyperfinite II$_1$-factor case, the number of projections that can be seen explicitly in any sense from the above picture is minimal. And even then, very often it is more convenient to see the hyperfinite II$_1$-factor under other presentations; typically, you take an appropriate group $G$ (ICC, and such that it is an incresing union of finite subgroups) and you get the hyperfinite II$_1$-factor as the sot-closure of the span of the image of $G$ under the left regular representation. In this picture, which as I say is often more useful than the matricial one, not a single projection can be written explicitly. Similarly when you construct your factor as a crossed product of a certain $L^\infty(X)$ and a group action on $X$.

The situation I describe above is just much worse in type III factors. In a type III factor any two nonzero projections are equivalent. So if you want to look at the "concrete" picture you have in UHF$(2^\infty)$, you have for example that the two projections $$ \begin{bmatrix} 1&0&0&0\\ 0&0&0&0\\0&0&0&0\\0&0&0&0\end{bmatrix} ,\qquad \begin{bmatrix} 1&0&0&0\\ 0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix} $$ are equivalent in the type III$_\lambda$ factor. You clearly won't get much mileage of having an "explicit" expression for those projections.

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  • $\begingroup$ Thanks for this great answer, Martin! You are right when you point abuse and misuse of the language in my question. Thanks for making it clear the difficulties regarding explicit expressions for projections. Since projections are so important in quantum theory, I wonder How vNA are used in QFT. Would you recommend any references for the introduction of the von Neumann algebras that can be obtained from UHF constructions? I’d also appreciate if you could share Introductory references for the study of the group measure space construction and other sources of von Neumann algebras. Regards. $\endgroup$
    – Lambda
    Oct 13, 2020 at 19:07
  • $\begingroup$ My main advance is to just plow through. As you get familiar with the topic you see how people use projecitons; often it's not an explicit expression what is useful. A very famous projection in von Neumann algebras is the Jones projection; it can be said explicitly what it is, but not in the way you imagine, I think. (continued) $\endgroup$ Oct 13, 2020 at 19:19
  • $\begingroup$ I don't know much if anything about QFT, so I cannot help you there. But if you look at the work of people like Doplicher, Longo, etc., etc., you could get a picture of what they do. (continued) $\endgroup$ Oct 13, 2020 at 19:20
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    $\begingroup$ The representations I mention are considered for instance (not an easy read) in Kadison-Ringrose section 12.3 (Theorem 12.3.8, concretely). Section 11.4 also contains related material (it seems to be more useful to consider UHF$(2^\infty)$ as an infinite tensor product rather than an inductive limit of matrix algebras). Crossed products are chapter 13 in the same book. $\endgroup$ Oct 13, 2020 at 19:25
  • $\begingroup$ I like the way things are presented in Sunder's book An Invitation to von Neumann Algebras. It has way less material than other sources, but it is clear, and closer to intuition than other books. I don't think a lot about von Neumann algebras these days, so these are the basic things that came up to mind. I'll mention some more references if I remember. $\endgroup$ Oct 13, 2020 at 19:27

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