# Preprints

A Inequality for Non-Microstates Free Entropy Dimension for Crossed Products by Finite Abelian Groups. |

Abstract: For certain generating sets of the subfactor pair M⊂M⋊G where G is a finite abelian group we prove an approximate inequality between their non-microstates free entropy dimension, resembling the Shreier formula for ranks of finite index subgroups of free groups. As an application, we give bounds on free entropy dimension of generating sets of crossed products of the form M⋊(Z/2Z)⊕∞ for a large class of algebras M |

Download: https://arxiv.org/abs/2201.09503 |

Duality for optimal couplings in free probability (with W. Gangbo, D. Jekel, K. Nam). |

Abstract: We study the free probabilistic analog of optimal couplings for the quadratic cost, where classical probability spaces are replaced by tracial von Neumann algebras, and probability measures on Rm are replaced by non-commutative laws of m-tuples. We prove an analog of the Monge-Kantorovich duality which characterizes optimal couplings of non-commutative laws with respect to Biane and Voiculescu's non-commutative L2-Wasserstein distance using a new type of convex functions. As a consequence, we show that if (X,Y) is a pair of optimally coupled m-tuples of non-commutative random variables in a tracial W∗-algebra A, then W∗((1−t)X+tY)=W∗(X,Y) for all t∈(0,1). Finally, we illustrate the subtleties of non-commutative optimal couplings through connections with results in quantum information theory and operator algebras. For instance, two non-commutative laws that can be realized in finite-dimensional algebras may still require an infinite-dimensional algebra to optimally couple. Moreover, the space of non-commutative laws of m-tuples is not separable with respect to the Wasserstein distance for m>1. |

Download: https://arxiv.org/abs/2105.12351 |

Tracial smooth functions of non-commuting variables and the free Wasserstein manifold (with D. Jekel, W. Li). |

Abstract: We formulate a free probabilistic analog of the Wasserstein manifold on Rd (the formal Riemannian manifold of smooth probability densities on Rd), and we use it to study smooth non-commutative transport of measure. The points of the free Wasserstein manifold W(R∗d) are smooth tracial non-commutative functions V with quadratic growth at ∞, which correspond to minus the log-density in the classical setting. The space of smooth tracial non-commutative functions used here is a new one whose definition and basic properties we develop in the paper; they are scalar-valued functions of self-adjoint d-tuples from arbitrary tracial von Neumann algebras that can be approximated by trace polynomials. The space of non-commutative diffeomorphisms D(R∗d) acts on W(R∗d) by transport, and the basic relationship between tangent vectors for D(R∗d) and tangent vectors for W(R∗d) is described using the Laplacian LV associated to V and its pseudo-inverse ΨV (when defined). Following similar arguments to arXiv:1204.2182, arXiv:1701.00132, and arXiv:1906.10051 in the new setting, we give a rigorous proof for the existence of smooth transport along any path t↦Vt when V is sufficiently close (1/2)∑jtr(x2j), as well as smooth triangular transport. |

Download: https://arxiv.org/abs/2101.06572 |

Fractional free convolution powers (with T. Tao (with appendix by D. Jekel)). |

Abstract: The extension k↦μ⊞k of the concept of a free convolution power to the case of non-integer k≥1 was introduced by Bercovici-Voiculescu and Nica-Speicher, and related to the minor process in random matrix theory. In this paper we give two proofs of the monotonicity of the free entropy and free Fisher information of the (normalized) free convolution power in this continuous setting, and also establish an intriguing variational description of this process |

Download: https://arxiv.org/abs/2009.01882 |

Cocycle Conjugacy of Free Bogoljubov Actions of R (with J. Keneda). |

Abstract: We show that Bogoljubov actions of R on the free group factor L(F∞) associated to sums of infinite multiplicity trivial and certain mixing representations are cocycle conjugate if and only if the underlying representations are conjugate. |

Download: https://arxiv.org/abs/2006.00351 |

Classification of regular subalgebras of the hyperfinite II_1 factor (with S. Popa, S. Vaes). |

Abstract: We prove that the regular von Neumann subalgebras B of the hyperfinite II_1 factor R satisfying the condition B′∩R=Z(B) are completely classified (up to conjugacy by an automorphism of R) by the associated discrete measured groupoid G. We obtain a similar classification result for triple inclusions A⊂B⊂R, where A is a Cartan subalgebra in R and the intermediate von Neumann algebra B is regular in R. A key step in proving these results is to show the vanishing cohomology for the associated cocycle actions of G on B. We in fact prove two very general vanishing cohomology results for free cocycle actions of amenable discrete measured groupoids on arbitrary tracial von Neumann algebras B, resp. Cartan inclusions A⊂B. Our work provides a unified approach and generalizations to many known vanishing cohomology and classification results [CFW81], [O85], [ST84], [BG84], [FSZ88], [P18], etc |

Download: https://arxiv.org/abs/1811.06929 |

Representing interpolated free group factors as group factors (with S. Popa). |

Abstract: We construct a one parameter family of ICC groups {Gt}t>1, with the property that the group factor L(Gt) is isomorphic to the interpolated free group factor L(Ft):=L(F2)1/t−1√, ∀t. Moreover, the groups Gt have fixed cost t, are strongly treeable and freely generate any treeable ergodic equivalence relation of same cost. |

Download: https://arxiv.org/abs/1805.10707 |

Free transport for convex potentials (with Y. Dabrowski, A. Guionnet). |

Abstract: We construct non-commutative analogs of transport maps among free Gibbs state satisfying a certain convexity condition. Unlike previous constructions, our approach is non-perturbative in nature and thus can be used to construct transport maps between free Gibbs states associated to potentials which are far from quadratic, i.e., states which are far from the semicircle law. An essential technical ingredient in our approach is the extension of free stochastic analysis to non-commutative spaces of functions based on the Haagerup tensor product. |

Download: https://arxiv.org/abs/1701.00132 |

Classification of a family of non almost periodic free Araki-Woods factors (with C. Houdayer and S. Vaes). |

Abstract: We obtain a complete classification of a large class non almost periodic free Araki-Woods factors Γ(μ,m) up to isomorphism. We do this by showing that free Araki-Woods factors Γ(μ,m) arising from finite symmetric Borel measures on R whose atomic part is nonzero and not concentrated on {0} have the joint measure class of all convolution powers of the measure as an invariant. Our key technical result is a deformation/rigidity criterion for the unitary conjugacy of two faithful normal states. We use this to also deduce rigidity and classification theorems for free product von Neumann algebras. |

Download: https://arxiv.org/abs/1605.06057 |

Von Neumann Algebras of Sofic Groups with First L2 Betti Number Zero are Strongly 1-Bounded. |

Abstract: We show that if Γ is a finitely generated finitely presented sofic group with zero first L2 Betti number and containing an element of infinite order, then the von Neumann algebra L(Γ) is strongly 1-bounded in the sense of Jung. In particular, L(Γ)≇L(Λ) if Λ is any group with free entropy dimension >1, for example a free group. The key technical result is a short proof of an estimate of Jung using non-microstates entropy techniques. |

Download: http://arxiv.org/abs/1604.08606 |

Cohomology and L2-Betti numbers for subfactors and quasi-regular inclusions (with S.Popa and S. Vaes). |

Abstract: We introduce L2-Betti numbers, as well as a general homology and cohomology theory for the standard invariants of subfactors, through the associated quasi-regular symmetric enveloping inclusion of II_1 factors. We actually develop a (co)homology theory for arbitrary quasi-regular inclusions of von Neumann algebras. For crossed products by countable groups Gamma, we recover the ordinary (co)homology of Gamma. For Cartan subalgebras, we recover Gaboriau's L2-Betti numbers for the associated equivalence relation. We prove that the L2-Betti numbers vanish for amenable inclusions. We compute the L2-Betti numbers for the standard invariants of the Temperley-Lieb-Jones subfactors and for the Fuss-Catalan subfactors. |

Download: http://arxiv.org/abs/1511.07329 |

Regularity of Polynomials in Free Variables (with I. Charlesworth). |

Abstract: We show that the spectral measure of any non-commutative polynomial of a non-commutative n-tuple cannot have atoms if the free entropy dimension of that n-tuple is n (see also work of Mai, Speicher, and Weber). Under stronger assumptions on the n-tuple, we prove that the spectral measure is not singular, and measures of intervals surrounding any point may not decay slower than polynomially as a function of the interval's length. |

Download: http://xxx.lanl.gov/abs/1408.0580 |

Free probability of type B and asymptotics of finite-rank perturbations of random matrices. |

Abstract: We show that finite rank perturbations of certain random matrices fit in the framework of infinitesimal (type B) asymptotic freeness. This can be used to explain the appearance of free harmonic analysis (such as subordination functions appearing in additive free convolution) in computations of outlier eigenvalues in spectra of such matrices. |

Download: http://arxiv.org/abs/1509.08841 |

Free analysis and planar algebras (with S. Curran, Y. Dabrowski). |

Abstract: We study 2-cabled analogs of Voiculescu's trace and free Gibbs states on Jones planar algebras. These states are traces on a tower of graded algebras associated to a Jones planar algebra. Among our results is that, with a suitable definition, finiteness of free Fisher information for planar algebra traces implies that the associated tower of von Neumann algebras consists of factors, and that the standard invariant of the associated inclusion is exactly the original planar algebra. We also give conditions that imply that the associated von Neumann algebras are non-Γ non-L2 rigid factors. |

Download: http://xxx.lanl.gov/abs/1411.0268 |

Freely Independent Random Variables with Non-Atomic Distributions (with P. Skoufranis). |

Abstract: We examine the distributions of non-commutative polynomials of non-atomic, freely independent random variables. In particular, we obtain an analogue of the Strong Atiyah Conjecture for free groups thus proving that the measure of each atom of any n x n matricial polynomial of non-atomic, freely independent random variables is an integer multiple of 1/ n. In addition, we show that the Cauchy transform of the distribution of any matricial polynomial of freely independent semicircular variables is algebraic and thus the polynomial has a distribution that is real-analytic except at a finite number of points. |

Download: http://arxiv.org/abs/1305.1920 |

Free Monotone Transport (with A. Guionnet). |

Abstract: By solving a free analog of the Monge-Amp`ere equation, we prove a non-commutative analog of Brenier's monotone transport theorem: if an $n$-tuple of self-adjoint non-commutative random variables $Z_{1},...,Z_{n}$ satisfies a regularity condition (its conjugate variables $xi_{1},...,xi_{n}$ should be analytic in $Z_{1},...,Z_{n}$ and $xi_{j}$ should be close to $Z_{j}$ in a certain analytic norm), then there exist invertible non-commutative functions $F_{j}$ of an $n$-tuple of semicircular variables $S_{1},...,S_{n}$, so that $Z_{j}=F_{j}(S_{1},...,S_{n})$. Moreover, $F_{j}$ can be chosen to be monotone, in the sense that $F_{j}=mathscr{D}_{j}g$ and $g$ is a non-commutative function with a positive definite Hessian. In particular, we can deduce that $C^{*}(Z_{1},...,Z_{n})cong C^{*}(S_{1},...,S_{n})$ and $W^{*}(Z_{1},...,Z_{n})cong L(mathbb{F}(n))$. Thus our condition is a useful way to recognize when an $n$-tuple of operators generate a free group factor. We obtain as a consequence that the q-deformed free group factors $Gamma_{q}(mathbb{R}^{n})$ are isomorphic (for sufficiently small $q$, with bound depending on $n$) to free group factors. We also partially prove a conjecture of Voiculescu by showing that free Gibbs states which are small perturbations of a semicircle law generate free group factors. Lastly, we show that entrywise monotone transport maps for certain Gibbs measure on matrices are well-approximated by the matricial transport maps given by free monotone transport. |

Download: http://arxiv.org/abs/1204.2182 |

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 II_{1} 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 II_{1} factor M containing an ''exotic'' maximal abelian subalgebra A: as an A,A-bimodule, L^{2}(M) is neither coarse nor discrete. Thus we show that there exist II_{1} 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 |