arXiv subjects

David Jekel

Publications and source records attributed to David Jekel.

At least 19 recordsLinked to original sources

arXiv2026

Compact quantum metric spaces from free probability

We study quantum metric space structures on operator algebras arising from free probability, namely those associated to $q$-Gaussians and free Gibbs laws for convex potentials. We note that even for free semicirculars, Voiculescu's dual system does not produce a quantum metric space structure that recovers the weak-$*$ topology on the state space. However, for $q$-Gaussians, we can define a compact quantum metric space using length-like functions by the same method as has already been used for hyperbolic groups, quantum groups of rapid decay, free products, and free graph algebras. Next, motivated by the free transport results for free Gibbs laws, we describe a universal way of defining Lip-norms in terms of a generating set, which behaves well under changes of coordinates. We show using semigroup regularization that this Lip-norm defines a quantum metric space structure for $q$-Gaussians, and then transfer this property to free Gibbs laws for convex potentials using free transport.

math.OA↗

arXiv2026

Simplicity of q-Gaussian C*-algebras

We show that q-Gaussian $C^*$-algebras for $q \in (-1, 1)$ have the Dixmier averaging property, and hence are simple with a unique trace. We argue by combining rapid decay and spectral gap estimates for the commutators with the generators, which are obtained from free probability.

math.OA↗

arXiv2026

Questions on the structure of random embeddings of $L(\mathbb{F}_2)$

Motivated by recent developments at the interface of operator algebras and random matrix theory, we propose new conjectures concerning the asymptotic structure of random matrix models of the countable free groups. The first conjecture predicts a random matrix analogue of the Akemann-Ostrand property for free groups, and reveals a succinct approach to recover the Peterson-Thom property for $L(\mathbb{F}_2)$. The second stronger conjecture is motivated by continuous model theory. It predicts that the \emph{random} embedding of the free group factor into a matrix ultraproduct is \emph{existential}. We discuss the interesting relationship between these conjectures.

math.OA↗

arXiv2026

Free information geometry and the model theory of noncommutative stochastic processes

We study entropy and optimal transport theory in the free probabilistic setting motivated by the large-$n$ theory of random tuples of matrices. We define a new version of free entropy $\chi_{\operatorname{chron}}^{\mathcal{U}}$, which is concave along geodesics in the corresponding Wasserstein space. Moreover, the heat evolution satisfies the evolution variational inequality, which means that the heat evolution is the Wasserstein gradient flow for entropy in the metric sense. It also has further desirable properties such as a chain rule for iterated conditioning, and an expression in terms of stochastic control problems. This entropy is defined using microstate spaces of matrix approximations with respect to an expanded class of test functions called chronological formulas, which are constructed so as to be closed under taking partial suprema and infima and application of a free heat semigroup. These formulas are part of a novel framework for studying noncommutative filtrations and stochastic processes as metric structures in the sense of continuous model theory.

math.OA↗

arXiv2026

Asymptotic expansion for transport maps between laws of multimatrix models

We study the large-$N$ behavior of random matrix tuples $Y^N = (Y_1^N,\dots,Y_d^N)$ with joint density proportional to $e^{-N^2 V}$ for some convex function $V$ in non-commuting variables satisfying certain bounds on its second derivative. We give an asymptotic expansion in powers of $1/N^2$ of the trace of noncommutative smooth functions of $Y^N$. We also give an asymptotic expansion for a family of maps $T^N$ that transport the law of a tuple of independent GUE random matrices to the law of $Y^N$ and, as a consequence, show strong convergence for the multimatrix models $Y^N$. Our proof is based on an asymptotic expansion for the heat semigroup associated to the measure, which is expressed in terms of smooth functions of a matrix Brownian motion $(S^{N}_t)_{t \geq 0}$. We introduce spaces of noncommutative smooth functions that unify and generalize the cases of polynomials and single-variable smooth functions and allow the systematic application of asymptotic expansion techniques to multimatrix models with convex interaction.

math.PR↗

arXiv2026

Exotic full factors via weakly coarse bimodules

We are able to explicitly compute the bimodule structure of von Neumann algebra inclusions in handle constructions, which arise as inductive limits of iterated amalgamated free products not elementarily equivalent to $L(\mathbb{F}_2)$. Our computation is achieved via identifying delicate normal form decompositions in amalgamated free products built in an iterated fashion. Using these techniques, we are able to show that the handles constructions are always full, without any need to appeal to Property (T) phenomena which was essential in all previous works. Furthermore our bimodule machinery works in the setting of arbitrary von Neumann algebras equipped with faithful normal states, yielding examples of full $\mathrm{III}_1$ factors via handle constructions.

math.OA↗

arXiv2026

Bigraph independence : a mixture of the five natural independences

We introduce a notion of non-commutative joint independence for multiple algebras in a non-commutative probability space. The pairwise relationships between these algebras are encoded by a graph with two edge sets -- a combinatorial structure we call a bigraph -- and naturally encompass the five fundamental types of independence: tensor, free, (anti)monotone, and Boolean. It subsumes the BMT independence of Arizmendi--Mendoza--Vazquez-Becerra (when all pairwise relationships are Boolean, (anti)monotone, or tensor) and the $\epsilon$ or $\Lambda$-independence of Mlotkowski (when the pairwise relationships are tensor and free). We present explicit combinatorial moment formulas, a Hilbert space construction, and natural associativity relations within this setting. Furthermore, we demonstrate that bigraph independence emerges in the asymptotic behavior of tensor product random matrix models with respect to a vector state, encompassing the Charlesworth--Collins model for $\varepsilon$-independence as a special case and offering a random matrix perspective on BMT independence.

math.PR↗

arXiv2025

Elementary embeddings into ultrapower $\mathrm{II}_1$ factors without a UCP lift

We show that there are $\mathrm{II}_1$ factors $M$ and elementary embeddings $M \to M^{\mathcal{U}}$ which do not lift to sequences of UCP maps, and in fact $M$ can be chosen from any given elementary equivalence class. Furthermore, under continuum hypothesis, we show that in the sense of cardinality "most" automorphisms of a ultrapower $M^{\mathcal{U}}$ of a separable $\mathrm{II}_1$ factor do not lift to a sequence of UCP maps $\varphi_n: M \to M$.

math.OA↗

arXiv2025

Large $n$-limit of matrix control problems and non-commutative controls

Building on the free-probability stochastic control framework introduced in arXiv:2502.17329, we connect optimal control problems for $n \times n$ random matrix ensembles with their infinite-dimensional, free-probability analogues. Under natural convexity hypotheses, we prove that the non-commutative value function captures the large-$n$ limit of the corresponding finite-matrix control problems. As an application, we give a new perspective on the Laplace principle for convex functionals in the theory of large deviations for random matrices.

math.AP↗

arXiv2025

The unitary group of a $\mathrm{II}_1$ factor is SOT-contractible

We show that the unitary group of any SOT-separable $\mathrm{II}_1$ factor $M$, with the strong operator topology, is contractible. Combined with several old results, this implies that the same is true for any SOT-separable von Neumann algebra with no type $\mathrm{I}_n$ direct summands ($n < \infty$). The proof for the $\mathrm{II}_1$-factor case uses regularization via free convolution and Popa's theorem on the existence of approximately free Haar unitaries in $\mathrm{II}_1$ factors.

math.OA↗

arXiv2025

Strong convergence to operator-valued semicirculars

We establish a framework for weak and strong convergence of matrix models to operator-valued semicircular systems parametrized by operator-valued covariance matrices $\eta = (\eta_{i,j})_{i,j \in I}$. Non-commutative polynomials are replaced by covariance polynomials that can involve iterated applications of $\eta_{i,j}$, leading to the notion of covariance laws. We give sufficient conditions for weak and strong convergence of general Gaussian random matrices and deterministic matrices to a $B$-valued semicircular family and generators of the base algebra $B$. In particular, we obtain operator-valued strong convergence for continuously weighted Gaussian Wigner matrices, such as Gaussian band matrices with a continuous cutoff, and we construct natural strongly convergent matrix models for interpolated free group factors.

math.OA↗

arXiv2025

The atoms of graph product von Neumann algebras

We completely classify the atomic summands in a graph product $(M,\varphi) = *_{v \in \mathcal{G}} (M_v,\varphi_v)$ of von Neumann algebras with faithful normal states. Each type I factor summand $(N,\psi)$ is a tensor product of type I factor summands $(N_v,\psi_v)$ in the individual algebras. The existence of such a summand and its weight in the direct sum can be determined from the $(N_v,\psi_v)$'s using explicit polynomials associated to the graph.

math.OA↗

arXiv2025

Quantum Wasserstein distances for quantum permutation groups

We seek an analog for the quantum permutation group $S_n^+$ of the normalized Hamming distance for permutations. We define three distances on the tracial state space of $C(S_n^+)$ that generalize the $L^1$-Wasserstein distance of probability measures on $S_n$ equipped with the normalized Hamming metric, for which we demonstrate basic metric properties, subadditivity under convolution, and density of the Lipschitz elements in the $\mathrm{C}^{\ast}$-algebra.

math.OA↗

arXiv2025

Viscosity Solutions in Non-commutative Variables

Motivated by parallels between mean field games and random matrix theory, we develop stochastic optimal control problems and viscosity solutions to Hamilton-Jacobi equations in the setting of non-commutative variables. Rather than real vectors, the inputs to the equation are tuples of self-adjoint operators from a tracial von Neumann algebra. The individual noise from mean field games is replaced by a free semi-circular Brownian motion, which describes the large-$n$ limit of Brownian motion on the space of self-adjoint matrices. We introduce a classical common noise from mean field games into the non-commutative setting as well, allowing the problems to combine both classical and non-commutative randomness.

math.AP↗

arXiv2025

Operator models and analytic subordination for operator-valued free convolution powers

We revisit the theory of operator-valued free convolution powers given by a completely positive map $\eta$. We first give a general result, with a new analytic proof, that the $\eta$-convolution power of the law of $X$ is realized by $V^*XV$ for any operator $V$ satisfying certain conditions, which unifies Nica and Speicher's construction in the scalar-valued setting and Shlyakhtenko's construction in the operator-valued setting. Second, we provide an analog, for the setting of $\eta$-valued convolution powers, of the analytic subordination for conditional expectations that holds for additive free convolution. Finally, we describe a Hilbert-space manipulation that explains the equivalence between the $n$-fold additive free convolution and the convolution power with respect to $\eta = n \operatorname{id}$.

math.OA↗

arXiv2025

Information geometry for types in the large-$n$ limit of random matrices

We study the interaction between entropy and Wasserstein distance in free probability theory. In particular, we give lower bounds for several versions of free entropy dimension along Wasserstein geodesics, as well as study their topological properties with respect to Wasserstein distance. We also study moment measures in the multivariate free setting, showing the existence and uniqueness of solutions for a regularized version of Santambrogio's variational problem. The role of probability distributions in these results is played by types, functionals which assign values not only to polynomial test functions, but to all real-valued logical formulas built from them using suprema and infima. We give an explicit counterexample showing that in the framework of non-commutative laws, the usual notion of probability distributions using only non-commutative polynomial test functions, one cannot obtain the desired large-$n$ limiting behavior for both Wasserstein distance and entropy simultaneously in random multi-matrix models.

math.OA↗

arXiv2024

Elementary equivalence and disintegration of tracial von Neumann algebras

We prove an analog of the disintegration theorem for tracial von Neumann algebras in the setting of elementary equivalence rather than isomorphism, showing that elementary equivalence of two direct integrals implies fiberwise elementary equivalence under mild, and necessary, hypotheses. This verifies a conjecture of Farah and Ghasemi. Our argument uses a continuous analog of ultraproducts where an ultrafilter on a discrete index set is replaced by a character on a commutative von Neumann algebra, which is closely related to Keisler randomizations of metric structures. We extend several essential results on ultraproducts, such as {\L}o\'s's theorem and countable saturation, to this more general setting.

math.OA↗

arXiv2024

General solidity phenomena and anticoarse spaces for type $\mathrm{III}_1$ factors

By developing a theory of anticoarse spaces in the purely infinite setting and using 1-bounded entropy techniques along with recent strong convergence results in random matrix theory, we show that free Araki--Woods factors offer the first examples of type $\mathrm{III}$ factors satisfying vastly general degrees of indecomposability phenomena. Notably this includes strong solidity with respect to any weakening of the normalizer currently in the literature and the Peterson--Thom property.

math.OA↗