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.

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