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Dimitri Shlyakhtenko

Publications and source records attributed to Dimitri Shlyakhtenko.

At least 19 recordsLinked to original sources
arXiv2026

A Free Analog of Bobkov's Gaussian Isoperimetry Inequality

We prove a one-variable functional inequality which is the free probability analog of Bobkov's isoperimetry inequality. The inequality involves the $L^1$ norm of the difference quotient of a function $f$ and can be viewed as a non-local isoperimetric inequality. We also prove related inequalities for subsets of an interval as well as for subsets of roots of Hermite polynomials. This paper is also an experiment in AI-based exploration of free analogs of classical probability statements.

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arXiv2026

Failure of Strong Convergence of Matrices with Fermionic Entries

Let $Q^{(k)}_N$ be an $N\times N$ matrices with entries satisfying CAR, normalized to have variance $1/\sqrt{N}$ with respect to the trace of the CAR algebra. We show that, although the operator norm of the real part of an individual matrix $Q^{(k)}_N$ converges as $N\to\infty$ to the semicircular limit, the family of matrices does not converge to the free probability limit strongly. In fact, even the operator space structure of the linear spans of the real and imaginary parts of $Q^{(k)}_N$'s, $k=1,\dots,M$, does not converge to the semicircular limit.

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arXiv2025

Establishing strong 1-boundedness via non-microstates free entropy techniques

We show that, for many choices of finite tuples of generators $X = (x_1, \dots , x_d)$ of a tracial von Neumann algebra $(M, τ)$ satisfying certain decomposition properties (non-primeness, possessing a Cartan subalgebra, or property $Γ$), one can find a diffuse, hyperfinite subalgebra $N \subseteq (W^*(X))^ω$ (often in $W^*(X)$ itself), such that $W^*(N,X+\sqrt{t}S) = W^*(N,X,S)$ for all $t > 0$. (Here $S$ is a free semicircular family, free from $\{X\} \cup N$). This gives a short non-microstates proof of strong 1-boundedness for such algebras.

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arXiv2022

Duality for optimal couplings in free probability

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 $\mathbb{R}^m$ 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 $L^2$-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 $\mathrm{W}^*$-algebra $\mathcal{A}$, then $\mathrm{W}^*((1 - t)X + tY) = \mathrm{W}^*(X,Y)$ for all $t \in (0,1)$. 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 $m > 1$.

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arXiv2021

Tracial smooth functions of non-commuting variables and the free Wasserstein manifold

We formulate a free probabilistic analog of the Wasserstein manifold on $\mathbb{R}^d$ (the formal Riemannian manifold of smooth probability densities on $\mathbb{R}^d$), and we use it to study smooth non-commutative transport of measure. The points of the free Wasserstein manifold $\mathscr{W}(\mathbb{R}^{*d})$ are smooth tracial non-commutative functions $V$ with quadratic growth at $\infty$, 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 $\mathscr{D}(\mathbb{R}^{*d})$ acts on $\mathscr{W}(\mathbb{R}^{*d})$ by transport, and the basic relationship between tangent vectors for $\mathscr{D}(\mathbb{R}^{*d})$ and tangent vectors for $\mathscr{W}(\mathbb{R}^{*d})$ is described using the Laplacian $L_V$ associated to $V$ and its pseudo-inverse $Ψ_V$ (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 $t \mapsto V_t$ when $V$ is sufficiently close $(1/2) \sum_j \operatorname{tr}(x_j^2)$, as well as smooth triangular transport.

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arXiv2021

Fractional free convolution powers

The extension $k \mapsto μ^{\boxplus k}$ of the concept of a free convolution power to the case of non-integer $k \geq 1$ 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.

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arXiv2020

Cocycle Conjugacy of Free Bogoljubov Actions of $\mathbb{R}$

We show that Bogoljubov actions of $\mathbb{R}$ on the free group factor $L(\mathbb{F}_{\infty})$ associated to sums of infinite multiplicity trivial and certain mixing representations are cocycle conjugate if and only if the underlying representations are conjugate.

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arXiv2019

Classification of regular subalgebras of the hyperfinite II_1 factor

We prove that the regular von Neumann subalgebras $B$ of the hyperfinite II_1 factor $R$ satisfying the condition $B'\cap R=Z(B)$ 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 $A\subset B \subset R$, 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 $A\subset B$. Our work provides a unified approach and generalizations to many known vanishing cohomology and classification results [CFW81], [O85], [ST84], [BG84], [FSZ88], [P18], etc.

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arXiv2018

Representing interpolated free group factors as group factors

We construct a one parameter family of ICC groups $\{G_t\}_{t>1}$, with the property that the group factor $L(G_t)$ is isomorphic to the interpolated free group factor $L(\mathbb F_t):=L(\mathbb{F}_2)^{1/\sqrt{t-1}}$, $\forall t$. Moreover, the groups $G_t$ have fixed cost $t$, are strongly treeable and freely generate any treeable ergodic equivalence relation of same cost.

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arXiv2017

Classification of a family of non almost periodic free Araki-Woods factors

We obtain a complete classification of a large class of 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 $μ$ on $\mathbf{R}$ whose atomic part $μ_a$ is nonzero and not concentrated on $\{0\}$ have the joint measure class $\mathcal C(\bigvee_{k \geq 1} μ^{\ast k})$ 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.

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arXiv2016

Cohomology and $L^2$-Betti numbers for subfactors and quasi-regular inclusions

We introduce $L^2$-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 $Γ$, we recover the ordinary (co)homology of $Γ$. For Cartan subalgebras, we recover Gaboriau's $L^2$-Betti numbers for the associated equivalence relation. In this common framework, we prove that the $L^2$-Betti numbers vanish for amenable inclusions and we give cohomological characterizations of property (T), the Haagerup property and amenability. We compute the $L^2$-Betti numbers for the standard invariants of the Temperley-Lieb-Jones subfactors and of the Fuss-Catalan subfactors, as well as for free products and tensor products.

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arXiv2013

Freely Independent Random Variables with Non-Atomic Distributions

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 $n \times n$ matricial polynomial of non-atomic, freely independent random variables is an integer multiple of $n^{-1}$. 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.

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arXiv2011

On the symmetric enveloping algebra of planar algebra subfactors

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.

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arXiv2011

Strongly solid ${\rm II_1}$ factors with an exotic MASA

Using an extension of techniques of Ozawa and Popa, we give an example of a non-amenable strongly solid $\rm{II}_1$ 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 $\rm{II}_1$ 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.

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arXiv2006

All generating sets of all property T von Neumann algebras have free entropy dimension $\leq 1$

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, $δ_0(X)$, is less than or equal to 1. It follows that when N embeds into the ultraproduct of the hyperfinite $\mathrm{II}_1$-factor, then $δ_0(X)=1$ and otherwise, $δ_0(X)=-\infinity$. 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.

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arXiv2005

Remarks on free entropy dimension

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 factors.

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arXiv2005

Notes on Free Probability Theory

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.

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