# The Free Field: Realization via Unbounded Operators and Atiyah Property

Tobias Mai, Sheng Yin and myself have just uploaded our paper The free field: realizations via unbounded operators and Atiyah property to the archive. This is a new version of an older paper with similar title. There are quite a couple of changes compared to the previous version. First, we have cut out the parts related to absolute continuity (they will become part of another paper) and concentrate now on items which are mentioned in the title. Furthermore, what was before an implication in one direction, has now become an equivalence; however, for this we had to shift our attention from free entropy dimension $\delta^*$ to a related quantity $\Delta$.

Let me say a few words what this paper is about. Usually, in free probability, we are trying to understand the von Neumann algebra generated by some operators $X_1,\dots,X_n$. This is a quite tough question, to which we have, unfortunately, nothing to say for now. So, instead, we shift here somehow the perspective; by not looking on what we can generate out of our operators by taking analytic closures in the bounded operators, but instead looking on how far we can go with just an algebraic closure – however, by also allowing to take inverses. Of course, if we want to invert operators we are leaving quickly the bounded operators, so in order to get some nice class of objects, we consider this question within the unbounded operators. In general, unbounded operators are nasty, but luckily enough for us, we are usually in a tracial frame, where the von Neumann algebra generated by $X_1,\dots, X_n$ is type II $_1$, and in such a situation the affiliated unbounded operators are a much nicer class. In particular, they form an algebra and, even better, any operator there can be inverted if and only if it has no non-trivial kernel. In a more algebraic formulation: such an operator X has an inverse if and only if it does not have a zero divisor (in the corresponding von Neumann algebra).

So we ask now the question: what is the division closure of $X_1,\dots, X_n$ in the algebra of unbounded operators? The division closure is, by definition, the smallest algebra which contains $X_1,\dots, X_n$ and which is closed under taking inverses in case they exist as unbounded operators.

How nice can such a division closure be? The best we can expect is that it is actually a division ring (aka skew field), which means that every non-zero operator is invertible (which according to the above means that every non-zero operator has no non-trivial kernel). Note that usually we consider operators which are algebraically free, i.e., there are no polynomial relations between the $X_1,\dots, X_n$. This does, however, not exclude rational relations (i.e., relations which also involve inverses). If there are also no non-trivial rational relations then we get the so-called “free field” (actually, the “free skew field”). For example, if we have three free semicircular elements, then they satisfy neither non-trivial polynomial nor non-trivial rational relations, and the division closure in this case is the free field in three generators. On the other hand, for two free semicircular elements $X$ and $Y$ let us (as suggested by Ken Dykema and James Pascoe) consider $A:=Y^2$, $B:=YXY$, $C:=YX^2Y$. Then $A,B,C$ satisfy no polynomial relation, hence the algebra generated by them is the free algebra in 3 generators. However, they satisfy the non-trivial rational relation $BA^{-1}B-C=0$, and their division closure is a division ring, but not the free field (but a “localisation” of the free field).

The statements from the last paragraph, whether we get a skew field and whether this skew field is the free field, are quite non-trivial; and the main results from our paper are to provide tools for deciding this. Let me give a short version of two of the main results:

• the division closure is a division ring if and only if the operators $X_1,\dots, X_n$ satisfy the strong Atiyah property; the latter was introduced by Shlyakhtenko and Skoufranis (as an extension of the corresponding property from the group case);
• the division closure of $X_1,\dots, X_n$ is the free field if and only if $\Delta(X_1,\dots, X_n)$ is maximal (i.e., equal to n).

The quantity $\Delta$ was introduced by Connes and Shlyakhtenko in the context of their investigations on $L^2$ homology of von Neumann algebras. More precisely, $\Delta^*(X_1,\dots, X_n)=n$ (i.e., $X_1,\dots,X_n$ generate the free skew field) if and only if there exist no non-zero finite rank operators $T_1,\dots,T_n$ on $L^2(X_1,\dots,X_n)$ such that $\sum_i[T_i,X_i]=0$.

This maximality of $\Delta$ might not look very intuitive, so it is good that we can provide also some more useful sufficient criteria to ensure this. In particular, we have that $\Delta^*(X_1,\dots, X_n)=n$ if

• $\delta^*(X_1,\dots,X_n)=n$, where $\delta^*$ is the free entropy dimension; and we know many situations where this happens, like for free operators, where each $X_i$ is selfadjoint and has non-atomic distribution; this is for example the case for free semicirculars;
• $X_1,\dots,X_n$ has a dual system, i.e., operators $D_1,\dots,D_n$ on $L^2(X_1,\dots,X_n)$ such that $[X_i,D_j]=\delta_{i,j} P$, where $P$ is the projection onto the trace vector

As mentioned above, in the first version of our paper we had the implication that maximality of $\delta^*$ implies that our operators realize the free field. It took us a while to realize (and even more, to prove) that we can also go the other way, if we use $\Delta$ instead of $\delta^*$. It is actually not clear how far those quantities are from each other.

Let me also point out that in the original version we could only deal with selfadjoint operators; mainly, because $\delta^*$ makes only sense in such a setting. Working with $\Delta$ instead opened also the way to deal with the general situation. This allows in particular to recover in our setting also the old result of Linnell that the generators of the free group in the left regular representation generate the free field. Since those generators are not selfadjoint (but unitary), we needed to free our theory from the assumption of selfadjointness.

Finally, let me also mention that though all this looks quite abstract and algebraic it has also quite some consequences for the distribution of operators and, in particular, for the asymptotic eigenvalue distributions of random matrices. For this one has to realize that for selfadjoint functions in our operators the absence of a kernel means that the distribution has no atoms. Hence we can exclude atoms in the distributions of functions of our operators if they have maximal $\Delta$.

# “Non-Commutative Distributions”, Summer Term 2019

My class on “Non-commutative distributions” started today. The first lecture is already online, see our video platform. Actually, we have a new video system, so the sound should now be better than last term. I am not sure, though, whether this also applies to the frames.

The class will run during our summer term, which will end mid July. Since I will travel quite a bit during term, there will be some cancellations and reschedulings of lectures; nevertheless, I still hope that we will have in the end again something like 25 lectures.

The general topic of the class is progress which was made in the last couple of years on non-commutative distributions, and which relies on advances in

• the operator-valued version of free probability theory (in particular, for its analytic description)
• free analysis or free non-commutative function theory
• relating analytic questions about operators in von Neumann algebras with the theory (of Cohn et al.) of non-commutative linear algebra or the free skew field (aka as non-commutative rational functions)
• using the linearization trick to relate non-linear scalar problems with linear operator-valued problems

All of the above will be explained in the lectures. So don’t worry if you have no idea what all this actually means.

Much of this progress was actually achieved in recent years in the context of my ERC-Advanced Grant on “Non-Commutative Distributions”. As this grant has finished now, the class can also be seen as kind of final report for this.

I will assume some familiarity with basic functional analysis and complex analysis. It is surely also helpful to know at least a bit about free probability theory, but this can also be acquired by watching along the way a few of the videos from last term or reading relevant parts of the corresponding class notes.

# Free Activities in Montreal, Week 4 and Conclusion

## Week 4

The second workshop “Free Probability: the applied perspective”, in the last week of the program, had its focus on more applied directions. In recent years, it has become apparent that free probability, with its tools for calculating the eigenvalue distribution of random matrices or dealing with outliers, might have quite some impact for questions in applied subjects like: physics, statistics, wireless communications, or machine learning. Many of the talks during the second workshop addressed such issues and showed this surprising arch of free probability from the abstract to the very applied: we saw three vignettes on free probability and statistics; heard about a new theory for sketching in linear regression; and learned about free probability for deep learning.

## Conclusion

Apart from the exciting and surprising new connections and directions of free probability, there were also a couple of talks which presented new important results per se, creating a lot of discussions and potentially new collaborations. Just to mention a few, the talks “The free field meets free probability theory” by Mai and “The atoms of the free additive convolution of two operator-valued distributions” by Belinschi gave qualitatively new results about one of the hard problems of the analytic theory of free probability, namely the absence or the possible occurrence of atoms for various functions in non-commuting variables. The talk “Traffic independence and freeness over the diagonal” by Cebron highlighted important progress in the theory of traffic independence, by relating the up to now quite combinatorial theory to analytic tools from operator-valued freeness, and thus opening a quite new direction for this important concept. Another impressive advance, on the notoriously difficult task of calculating Brown measure, was presented in the talk “The Brown measure of free multiplicative Brownian
motion” by Kemp.

I think that the whole program on free probability at the CRM was a big success. In particular, the attendance of many young researchers, often for the whole month or a substantial part of it, and the many lively discussions in the seminar rooms and halls of the Pavillon Aisenstadt, following the talks or signalling ongoing or new collaborations, showed clearly that the subject is still very vibrant and full of new ideas. I am quite optimistic about the future of free probability theory.

# Free Activities in Montreal, Week 3

Our one-month program on New Developments in Free Probability and Applications has started. After a one-week workshop on the more theoretical side, there will be now two weeks of introductions and survey talks, as well as seminar talks, before we finish with another workshop on the (potential) applications of free probability.

Here is the schedule of talks for week 3, featuring in particular two of the three Aisenstadt Chair talks of Alice Guionnet. The first of those is intended for a general public, see also the CRM website. The third Aisenstadt Chair talk will be the opening talk of the second workshop.

WEDNESDAY March 20

Pavillon Claire-McNicoll, Université de Montréal, Room Z-220

16:15 Alice Guionnet (Aisenstadt Chair 2019)
“Free probability and Random matrices”
Free probability is the natural framework to consider matrices with size going to infinity. Since this key remark was made by Voiculescu in the nineties, these two fields have enriched each other continuously. We will discuss a few of these fruitful crossovers. This talk will only require a general mathematical background.

THURSDAY March 21

Pavillon André-Aisenstadt, Université de Montréal, Room 5340

09:30 – 11:00 Mireille Capitaine
“Introduction to Outliers for Deformed Wigner Matrices”

14:30 – 15:30 Felix Leid
“Maps, Partitioned Permutations, and Free Probability”

16:00 – 17:00 Maxime Fevrier
“From conditional freeness to infinitesimal freeness”

FRIDAY March 22

Pavillon André-Aisenstadt, Université de Montréal, Room 1360

10:30 Alice Guionnet (Aisenstadt Chair 2019)
“Free probability and random matrices: Conjugate variables and the Dyson-Schwinger equations”
In this talk I will discuss some uses of integration by parts in free probability, random matrices and related topics. In particular I will show how it can be used to study the large dimension asymptotics in random matrices and tilings.

# Free Activities in Montreal, Week 2

Our one-month program on New Developments in Free Probability and Applications has started. After a one-week workshop on the more theoretical side, there will be now two weeks of introductions and survey talks, as well as seminar talks, before we finish with another workshop on the (potential) applications of free probability.

Here is the schedule of talks for this week; they will be held in room 5340, 5th floor, Andre-Aisenstadt building on the campus of the Universite de Montreal.

Note that the second talk on Friday has changed; Felix Leid’s talk has been moved to next week, Ben Hayes will talk instead.

TUESDAY March 12

9:30 – 12:30 Tobias Mai, Roland Speicher, and Sheng Yin
“Introduction to Regularity and the Free Field”

Lunch break

2:30 – 4:00 Hari Bercovici
“Introduction to outliers”

THURSDAY March 14

9:30 – 12:30 Benson Au, Guillaume Cébron, and Camille Male
“Introduction to Traffic Freeness”

Lunch break

2:30 – 3:30 Laura Maassen
“Group-theoretical quantum groups”

4:00 – 5:00 Simon Schmidt
“Quantum automorphism groups of finite graphs”

FRIDAY March 15

9:30 – 10:30 Josué Váquez Becerra
“Fluctuation moments induced by some asymptotically liberating unitary matrices”

11:00 – 12:00 Ben Hayes
“A random matrix approach to the Peterson-Thom conjecture”

# Update on “Free Probability” Class

The lectures on our free probability class are over now. All 26 lectures are uploaded on the video platform, and the scans of my handwritten course notes can be found here.

I plan to continue next term with a class on “Non-Commutative Distributions”; this will in particular cover the operator-valued version of free probability and its use for dealing with polynomials in free variables, as well as addressing regularity properties of the distribution of such polynomials. There will be more cool stuff, but I still have to think about details. Our summer term starts in April, then I will be back with more information.

The plan is to continue with the recording of the lectures. If you have any suggestions on how to improve on this, please let me know.

# Asymptotic Freeness of Wigner Matrices

When preparing my lectures for the asymptotic freeness of various random matrix ensembles I stumbled about the situation concerning Wigner matrices. We all know that Wigner matrices and deterministic matrices are asymptotically free, but all the proofs I am aware of are annoyingly complicated. Shouldn’t there be a nice and simple proof without too many technicalities?

As was said in the section “Open problems and possible future directions” of the report of the 2008 Banff workshop “Free Probability, Extensions, and Applications”: “Whereas engineers have no problems in applying asymptotic freeness results for unitarily invariant ensembles it has become apparent that they do not have the same confidence in the analogous results for Wigner matrices. The main reason for this is the lack of precise statements on this in the literature. This has to be remedied in the future.”

Actually, at that time there existed already some results in this direction in the paper On Certain Free Product Factors via an Extended Matrix Model from 1993 of Ken Dykema. There the asymptotic freeness between independent Wigner matrices and diagonal (or even more general: block diagonal) deterministic matrices had been shown. But the case of general deterministic matrices remained open. Taking on the challenge by the engineers, we were determined to write down nice and accessible proofs, also including the full deterministic case.

The result of this was that such statements and proofs were included in the book An Introduction to Random Matrices of Greg Anderson, Alice Guionnet, and Ofer Zeitouni on one side and in my book with Jamie on the other side. However, I have to admit what looked like an easy exercise to Jamie and me at the beginning turned out to me much more complicated. An intermediate outcome of this was my paper with Jamie on Sharp Bounds for Sums Associated to Graphs of Matrices, but even with this as a nice black box the final proof still required a couple of technical pages in our book.

So I would like to come back to the original challenge and want to see what we really know about the relation between Wigner matrices and deterministic matrices. What are the clean statements and what are nice proofs. The situation for Wigner matrices is also more complicated than for Gaussian matrices, as the real and imaginary part for Wigner matrices do not have to be independent, hence the complex situation cannot be directly reduced to the real one, and questions about the *-freeness of non-selfadjoint Wigner matrices is also not exactly the same as the freeness of selfadjoint ones. Of course, all is related and similar, but if one asks a concrete question, usually it is hard to find the answer exactly for this in the literature.

I hope to collect here information about what is out there in the literature on that problem, with the final goal of coming up with the cleanest statements and the simplest proofs. If you have any information or ideas in this context, please let me know.