Category Archives: General

Welcome to the Non-Commutative World!

About two weeks ago I posted with Tobias Mai on the archive the preprint “A Note on the Free and Cyclic Differential Calculus”. Here is what we say in the abstract:

In 2000, Voiculescu proved an algebraic characterization of cyclic gradients of noncommutative polynomials. We extend this remarkable result in two different directions: first, we obtain an analogous characterization of free gradients; second, we lift both of these results to Voiculescu’s fundamental framework of multivariable generalized difference quotient rings. For that purpose, we develop the concept of divergence operators, for both free and cyclic gradients, and study the associated (weak) grading and cyclic symmetrization operators, respectively. One the one hand, this puts a new complexion on the initial polynomial case, and on the other hand, it provides a uniform framework within which also other examples – such as a discrete version of the Ito stochastic integral – can be treated.

At the moment I am not in the mood to say more specifically about this preprint (maybe Tobias or I will do so later), but I want to take the opportunity — in particular as the first anniversary of this blog is also coming closer — to put this in a bigger context and mumble a bit about the bigger picture and our dreams … so actually about what this blog should be all about.

Free probability theory has come a long way. Whereas born in the subject of operator algebras, the realization that is also has to say quite a bit about random matrices paved the way to its use in many (and, in particular, also applied) subjects. Hence there are now also papers in statistics, like this one, or in deep learning, like this one or this one, which use tools from free probability for their problems. The last words on how far the use of free probability goes in those subjects are surely not yet spoken but I am looking forward to see more on this.

This is of course all great and nice for our subject, but on the other hand there is also a bigger picture in the background, where I would hope for some more fundamental uses of free probability.

This goes roughly like this. There is the classical world, where we are dealing with numbers and functions and everything commutes; then there is our non-commutative world, where we are dealing with operators and limits of random matrices and where on the basic level nothing commutes. That’s where quite a bit of maximal non-commutative mathematics has been (and is still being) developed from various points of views:

  • free probability deals with a non-commutative notion of independence for non-commuting random variables;
  • there is a version of a non-commutative differential calculus which allows to talk about derivatives in non-commutative variables; my paper with Tobias mentioned above is in this context and tries to formalize and put all this a bit further;
  • free analysis (or free/non-commutative function theory) aims at a non-commutative version of classical complex analysis, i.e., a theory of analytic functions in non-commuting variables;
  • free quantum groups provide the right kind of symmetries for such non-commuting variables.

The nice point is that all those subjects have their own source of motivation but it turns out that there are often relations between them which are non-commutative analogues of classical results.

So, again this is all great and nice, BUT apart from the commutative and our maximal non-commutative world there is actually the, maybe most important, quantum world. This is of course also non-commutative, but only up to some point. There operators don’t commute in general, but commutativity is replaced by some other relations, like the canonical commutation relations, and there are actually still a lot of operators which commute (for example, measurements which are at space-like positions are usually modeled by commuting operators). Because of this commutativity, basic concepts of free probability do not have a direct application there.

Here is a bit more concretely what I mean with that. In free probability we have free analogues of such basic concepts as entropy or Fisher information. There are a lot of nice statements and uses of those concepts and via random matrices they can also be seen as arising as a kind of large N limit of the corresponding classical concepts. However, in the classical world those concepts have usually also a kind of operational meaning by being the answer to fundamental questions. For example, the classical Shannon entropy is the answer to the question how much information one can transmit over classical channels. Now there are quantum channels and one can ask how much information one can transmit over them; again there are answers in terms of an entropy, but this is unfortunately not free entropy, but von Neumann entropy, a more commutative non-commutative cousin of classical entropy. There are just too many tensor products showing up in the quantum world which prevent a direct use of basic free probability concepts. But still, I am dreaming of finding some day operational meanings of free entropy and similar quantities.

Anyhow, I hope to continue to explain in this blog more of the concrete results and problems which we have in free probability and related subjects; but I just wanted to point out that there are also some bigger dreams in the background.

SYK Model and q-Brownian Motion

Recently I became aware of the so-called Sachdev-Ye-Kitaev (SYK) model, which has attracted quite some interest in the last couple of years in physics, as a kind of toy model for quantum holography. What attracted my attention was the fact that in some limit there appears a q-deformation of the Gauss distribution – the same one which also showed up in my old papers with Marek Bozejko and Burkhard K├╝mmerer on non-commutative versions of Brownian motions, see here and here. Whereas in the SYK context there is usually only one limit distribution, in our non-commutative probability context we usually have the multivariate situation with several random variables (corresponding to the increments of the process). Thus I wanted to see whether one can also extend the calculations in the SYK model to a multivariate setting. This is done together with Miguel Pluma in our paper The SYK Model and the q-Brownian Motion. It turns out that one gets indeed q-Gaussian variables corresponding to orthogonal vectors for independent SYK models.

It is not clear to me whether such independent copies of SYK models have any physical relevance. However, there have recently been some papers by Berkooz and collaborators, here and here, where they calculated the 2-point and the 4-point function for the large N double scaled SYK model, by using also essentially the combinatorics of such multivariate extensions.

Those calculations are quite technical and not easy, and it seems to be unclear whether one can get a final analytic result. This seems to be related to our problems of doing any useful analytic calculations with the multivariate q-Gaussian distribution, which is one of the main obstructions for progress on free entropy or Brown measures for the q-Gaussian distributions. (Okay, there has been some progress via free transport by Alice Guionnet and Dima Shlyakhtenko, but this is quite abstract without concrete analytic formulas.) It would be nice (and surely helpful) if we could get some more concrete description of the operator-valued Cauchy transform of the multivariate q-Gaussian distribution.

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.



Welcome to “Free Probability Theory”

Hello!

This is a blog on topics around Free Probability Theory. Originally, I created this to provide a forum around my lecture on free probability theory. But now I plan (at least hope) to extend it to general blog on free probability theory.

So I hope to post here also all kind of information which is relevant in the context of free probability theory, like: meetings, new results, discussions of open problems or general directions in the subject.

I hope that others will also make some contributions; if you are interested in writing your own posts in this context, please contact me!