Preprints

On operator-valued free convolution powers.
Abstract: We give an explicit realization of the eta-convolution power of an A-valued distribution, as defined earlier by Anshelevich, Belinschi, Fevrier and Nica. If eta is a completely positive map from A to A and eta is greater than the identity map, we give a short proof of positivity of the eta-convolution power of a positive distribution. Conversly, if the eta is not bigger than id, we construct an s-tuple whose A-valued distribution is positive, but has non-positive eta-convolution power.
Download: http://arxiv.org/abs/1110.5127
On the symmetric enveloping algebra of planar algebra subfactors (with S. Curran, V. Jones).
Abstract: We give a diagrammatic description of Popa's symmetric enveloping algebras associated to planar algebra subfactors. As an application we construct a natural family of derivations on these factors, and compute a certain free entropy dimension type quantity.
Download: http://arxiv.org/abs/1105.1721
Loop models, random matrices and planar algebras (with A. Guionnet, V.F.R. Jones, P. Zinn-Justin).
Abstract: We define matrix models that converge to the generating functions of a wide variety of loop models with fugacity taken in sets with an accumulation point. The latter can also be seen as moments of a non-commutative law on a subfactor planar algebra. We apply this construction to compute the generating functions of the Potts model on a random planar map.
Download: http://lanl.arxiv.org/abs/1012.0619
Free probability, Planar algebras, Subfactors and Random Matrices.
Abstract: To a planar algebra P in the sense of Jones we associate a natural non- commutative ring, which can be viewed as the ring of non-commutative polynomials in several indeterminates, invariant under a symmetry encoded by P. We show that this ring carries a natural structure of a non-commutative probability space. Non-commutative laws on this space turn out to describe random matrix ensembles possessing special sym- metries. As application, we give a canonical construction of a subfactor and its symmetric enveloping algebra associated to a given planar algebra P. This talk is based on joint work with A. Guionnet and V. Jones.
Download: http://lanl.arxiv.org/abs/1009.0939
A semi-finite algebra associated to a planar algebra (with A. Guionnet and V.F.R. Jones).
Abstract: We canonically associate to any planar algebra two type II_{infty} factors M_{+} and M_{-}. The subfactors constructed previously by the authors in a previous paper are isomorphic to compressions of M_{+} and M_{-} to finite projections. We show that each mathfrak{M}_{pm} is isomorphic to an amalgamated free product of type I von Neumann algebras with amalgamation over a fixed discrete type I von Neumann subalgebra. In the finite-depth case, existing results in the literature imply that M_{+} cong M_{-} is the amplification a free group factor on a finite number of generators. As an application, we show that the factors M_{j} constructed in our previous paper are isomorphic to interpolated free group factors L(mathbb{F}(r_{j})), r_{j}=1+2delta^{-2j}(delta-1)I, where delta^{2} is the index of the planar algebra and I is its global index. Other applications include computations of laws of Jones-Wenzl projections.
Download: http://arxiv.org/abs/0911.4728
Strongly solid II1 factors with an exotic MASA (with C. Houdayer).
Abstract: Using an extension of techniques of Ozawa and Popa, we give an example of a non-amenable strongly solid II1 factor M containing an ''exotic'' maximal abelian subalgebra A: as an A,A-bimodule, L2(M) is neither coarse nor discrete. Thus we show that there exist II1 factors with such property but without Cartan subalgebras. It also follows from Voiculescu's free entropy results that M is not an interpolated free group factor, yet it is strongly solid and has both the Haagerup property and the complete metric approximation property.
Download: http://arxiv.org/abs/0904.1225
Free probability of type B: analytic interpretation and applications (with S. T. Belinschi).
Abstract: In this paper we give an analytic interpretation of free convolution of type B, introduced by Biane, Goodman and Nica, and provide a new formula for its computation. This formula allows us to show that free additive convolution of type B is essentially a re-casting of conditionally free convolution. We put in evidence several aspects of this operation, the most significant being its apparition as an 'intertwiner' between derivation and free convolution of type A. We also show connections between several limit theorems in type A and type B free probability. Moreover, we show that the analytical picture fits very well with the idea of considering type B random variables as infinitesimal deformations to ordinary non-commutative random variables.
Download: http://arxiv.org/abs/0903.2721v1
An orthogonal approach to the subfactor of a planar algebra (with V. F. R. Jones and K. Walker).
Abstract: By changing to an orthogonal basis, we give a short proof that the subfactor of the graded algebra of a planar algebra reproduces the planar algebra.
Download: http://arxiv.org/abs/0807.4146
On Classical Analogues of Free Entropy Dimension (with A. Guionnet).
Abstract: We define a classical probability analogue of Voiculescu's free entropy dimension that we shall call the classical probability entropy dimension of a probability measure on $mathbb{R}^n$. We show that the classical probability entropy dimension of a measure is related with diverse other notions of dimension. First, it can be viewed as a kind of fractal dimension. Second, if one extends Bochner's inequalities to a measure by requiring that microstates around this measure asymptotically satisfy the classical Bochner's inequalities, then we show that the classical probability entropy dimension controls the rate of increase of optimal constants in Bochner's inequality for a measure regularized by convolution with the Gaussian law as the regularization is removed. We introduce a free analogue of the Bochner inequality and study the related free entropy dimension quantity. We show that it is greater or equal to the non-microstates free entropy dimension.
Download: http://lanl.arxiv.org/abs/math.PR/0701465
Free diffusions and Matrix models with strictly convex interaction (with A. Guionnet).
Abstract: We study solutions to the free stochastic differential equation $dX_t = dS_t - half DV(X_t)dt$, where $V$ is a locally convex polynomial potential in $m$ non-commuting variables. We show that for self-adjoint $V$, the law $mu_V$ of a stationary solution is the limit law of a random matrix model, in which an $m$-tuple of self-adjoint matrices are chosen according to the law $exp(-N textrm{Tr}(V(A_1,...,A_m)))dA_1... dA_m$. We show that if $V=V_beta$ depends on complex parameters $beta_1,...,beta_k$, then the law $mu_V$ is analytic in $beta$ at least for those $beta$ for which $V_beta$ is locally convex. In particular, this gives information on the region of convergence of the generating function for planar maps.

We show that the solution $dX_t$ has nice convergence properties with respect to the operator norm. This allows us to derive several properties of $C^*$ and $W^*$ algebras generated by an $m$-tuple with law $mu_V$. Among them is lack of projections, exactness, the Haagerup property, and embeddability into the ultrapower of the hyperfinite II$_1$ factor. We show that the microstates free entropy $chi(tau_V)$ is finite.

A corollary of these results is the fact that the support of the law of any self-adjoint polynomial in $X_1,...,X_n$ under the law $mu_V$ is connected, vastly generalizing the case of a single random matrix.

Download: http://lanl.arxiv.org/abs/math.OA/0701787
Random matrices, free probability, planar algebras and subfactors (with A. Guionnet and V.F.R. Jones).
Abstract: Using a family of graded algebra structures on a planar algebra and a family of traces coming from random matrix theory, we obtain a tower of non-commutative probability spaces, naturally associated to a given planar algebra. The associated von Neumann algebras are II$_{1}$ factors whose inclusions realize the given planar algebra as a system of higher relative commutants. We thus give an alternative proof to a result of Popa that every planar algebra can be realized by a subfactor.
Download: http://lanl.arxiv.org/abs/0712.2904
Lower estimates on microstates free entropy dimension.
Abstract: By proving that certain free stochastic differential equations have stationary solutions, we give a lower estimate on the microstates free entropy dimension of certain $n$-tuples $X_{1},...,X_{n}$: we show that $delta_{0}(X_{1},...,X_{n})geqdim_{Mbar{otimes}M^{o}}V$ where $M=W^{*}(X_{1},...,X_{n})$ and $V={(partial(X_{1}),...,partial(X_{n })):partialinmathcal{C}}$ is the set of values of derivations $A=mathbb{C}[X_{1},... X_{n}]to Aotime s A$ with the property that $partial^{*}partial(A)subset A$. Using similar techniques, we show that for $qgeq0$ sufficiently small (depending on $n$) and $X_{1},...,X_{n}$ a $q$-semicircular family, $delta_{0}(X_{1},...,X_{n})>1$. In particular, for small $qgeq0$, q-deformed free group factors have no Cartan subalgebras. An essential tool in our analysis is a free analog of an inequality between Wasserstein distance and Fisher information introduced by Otto and Villani (and also studied in the free case by Biane and Voiculescu).
Download: http://lanl.arxiv.org/abs/0710.4111
All generating sets of all property T von Neumann algebras have free entropy dimension <= 1 (with K. Jung).
Abstract: Suppose $N$ is a diffuse, property T von Neumann algebra and X is an arbitrary finite generating set of selfadjoint elements for N. By using rigidity/deformation arguments applied to representations of N in full matrix algebras, we deduce that the microstate spaces of X are asymptotically discrete up to unitary conjugacy. We use this description to show that the free entropy dimension of X, $delta_0(X)$, is less than or equal to 1. It follows that when N embeds into the ultraproduct of the hyperfinite $mathrm{II}_1$-factor, then $delta_0(X)=1$ and otherwise, $delta_0(X)=-infinity$. This generalizes the earlier results of Voiculescu, and Ge, Shen pertaining to $SL_n(mathbb Z)$ as well as the results of Connes, Shlyakhtenko pertaining to group generators of arbitrary property T algebras.
Download: http://front.math.ucdavis.edu/math.OA/0603669
A free analogue of Shannons problem on monotonicity of entropy.
Abstract: We prove a free probability analog of a result of Artstein-Ball-Barthe-Naor. In particualar we prove that if X_{1},X_{2},... are freely independent identically distributed random variables, then the free entropy chi(X_{1}+...+X_{n}/sqrt{n}) is monotone increasing for all n. Our proof also leads to a slight simplification of the original argument in the classical case.
Download: http://xxx.lanl.gov/abs/math.OA/0510103
Picard groups of topologically stable Poisson structures (with O. Radko).
Abstract: We compute the group of Morita self-equivalences (the Picard group) of a Poisson structure on an orientable surface, under the assumption that the degeneracies of the Poisson tensor are linear. The answer involves mapping class groups of surfaces, i.e., groups of isotopy classes of diffeomorphisms. We also show that the Picard group of these structures coincides with the group of outer Poisson automorphisms.
Download: http://xxx.lanl.gov/abs/math.SG/0408070
Notes on free probability theory.
Abstract: These notes are from a 4-lecture mini-course taught by the author at the conference on von Neumann algebras as part of the ``Geometrie non commutative en mathematiques et physique' month at CIRM in 2004.
Download: http://xxx.lanl.gov/abs/math.OA/0504063
Remarks on free entropy dimension.
Abstract: We prove a technical result, showing that the existence of a closable unbounded dual system in the sense of Voiculescu is equivalent to the finiteness of free Fisher information. This approach allows one to give a purely operator-algebraic proof of the computation of the non-microstates free entropy dimension for generators of groups carried out in an earlier joint work with I. Mineyev. The same technique also works for finite-dimensional algebras. We also show that Voiculescu's question of semi-continuity of free entropy dimension, as stated, admits a counterexample. We state a modified version of the question, which avoids the counterexample, but answering which in the affirmative would still imply the non-isomorphism of free group factor.
Download: http://xxx.lanl.gov/abs/math.OA/0504062
The microstates free entropy dimension of any DT-operator is 2 (with K. Dykema and K. Jung).
Abstract: Suppose that µ is an arbitrary Borel measure on the complex plane with compact support and take c > 0. If Z is a DT(µ,c)-operator as defined by Dykema and Haagerup, then the microstates free entropy dimension of Z is 2.
Download: http://xxx.lanl.gov/abs/math.OA/0412273
On multiplicity and free absorption for free Araki-Woods factors.
Abstract: We show that Ozawa's recent results on solid von Neumann algebras imply that there are free Araki-Woods factors, which fail to have free absorption. We also show that a free Araki-Woods factors $Gamma (mu, n)$ associated to a measure and a multiplicity function $n$ may non-trivially depend on the multiplicity function.
Download: http://xxx.lanl.gov/abs/math.OA/0302217