12d. Hyperfinite factors

[math] \newcommand{\mathds}{\mathbb}[/math]

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Back to general theory, there are many other things that can be said, in relation with hyperfiniteness. We first have a reduction theory result, as follows:

Theorem

Any tracial hyperfinite von Neumann algebra appears as

[[math]] A=\int_XA_x\,dx [[/math]]

with the factors [math]A_x[/math] being either usual matrix algebras, or the factor [math]R[/math].

Show Proof

This follows indeed by combining the von Neumann reduction theory from with the theory of [math]R[/math] of Murray-von Neumann and Connes .

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More generally, we have the following result, this time in arbitrary type:

Theorem

Given a hyperfinite von Neumann algebra [math]A\subset B(H)[/math], write its center as follows, with [math]X[/math] being a measured space:

[[math]] Z(A)=L^\infty(X) [[/math]]

The whole algebra [math]A[/math] decomposes then over this measured space [math]X[/math], as follows,

[[math]] A=\int_XA_x\,dx [[/math]]

with the fibers [math]A_x[/math] being hyperfinite von Neumann factors, which can be of type [math]{\rm I},{\rm II},{\rm III}[/math].

Show Proof

This is again something heavy, combining the general reduction theory results of von Neumann with the work of Connes in the hyperfinite case.

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Which brings us into the question of classifying all hyperfinite factors. The result here, due to Connes , with a key contribution by Haagerup , is as follows:

Theorem

The hyperfinite factors are as follows, with [math]1[/math] factor in each class

[[math]] {\rm I_N},{\rm I}_\infty [[/math]]

[[math]] {\rm II}_1,{\rm II}_\infty [[/math]]

[[math]] {\rm III}_0,{\rm III}_\lambda,{\rm III}_1 [[/math]]

and with the type [math]{\rm II}_1[/math] one [math]R[/math] being the most important, basically producing the others too.

Show Proof

This is again heavy, based on early work of Murray-von Neumann in type II , then on heavy work by Connes in type II and III , , basically finishing the classification, and with a final contribution by Haagerup in type [math]{\rm III}_1[/math] .

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Getting back now to the [math]{\rm II}_1[/math] factors, and beyond hyperfiniteness, where things are understood, with [math]R[/math] being the only example, there is a whole classification program here, by Popa and others, going on. Let us mention that a main open problem is that of deciding whether the free group factors are isomorphic or not: \begin{question} Are the von Neumann algebras of free groups isomorphic,

[[math]] L(F_N)\simeq L(F_M) [[/math]]

for [math]N\neq M[/math], or not? \end{question} This question can be of course asked in crossed product form, in the spirit of the various crossed product results evoked above, and of advanced ergodic theory in general, with the space in question, producing the crossed product, being the point:

[[math]] \{.\}\rtimes F_N\simeq^?\{.\}\rtimes F_M [[/math]]

This formulation, used by Popa, has the advantage of putting the above problem into a more conceptual framework, with lots of advanced machinery available around. However, it is not clear whether this formulation simplifies or not the original problem.

There are as well a number of alternative approaches to this question, and notably the Voiculescu one, using free probability, which is particularly conceptual and beautiful, the idea being that of recapturing the number [math]N\in\mathbb N[/math] from the knowledge of the von Neumann algebra [math]L(F_N)[/math], via an entropy-type invariant:

[[math]] L(F_N)\rightsquigarrow N [[/math]]

This latter program, while not solving the original problem, due to technical difficulties, is however very successful, in the sense that it has led to a lot of interesting results and computations, in relation with a lot of mathematics and physics.

Is the free group factor problem something belonging to logic, as the difficult problems in functional analysis usually end up being? No one really knows the answer here.

Interestingly, the question is difficult to the point where the conjectural answer, yes or no, is not known. And even worse, excluding the many people who have spent considerable time on the matter, years or more, working on yes or no, most people familiar with the question don't even really know what to wish for, yes or no, as an answer.

In what concerns us, we have been quite close in this book to the ideas of Voiculescu, but, as a surprise, these very ideas of Voiculescu lead us into wishing for a yes answer to the above question, which is opposite to his no wish, and work using free entropy. Indeed, to put things in context, let us formulate the question in the following way: \begin{question} Is there a factor [math]F[/math], standing as a free counterpart for [math]R[/math]? \end{question} And wouldn't you wish for a yes answer to this question, with [math]F[/math] being of course all the free group factors [math]L(F_N)[/math] combined, and probably many more, coming from all sorts of free quantum groups, free homogeneous spaces, or other free manifolds. It would be good to know in free geometry that what we get by default is this factor [math]F[/math].

As a last comment here, later on, when doing subfactors, we will see that the particular factor [math]F=L(F_\infty)[/math] quite does the job there, in subfactors, being more of less the only “free factor” that is needed, for that theory. But this does not really solve Question 12.31 in the context of subfactor theory because, ironically, the main questions there, including the “free” ones, rather concern the subfactors of the good old hyperfinite factor [math]R[/math].

General references

Banica, Teo (2024). "Principles of operator algebras". arXiv:2208.03600 [math.OA].