| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |
| 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 |