arXiv subjects

Yoann Dabrowski

Publications and source records attributed to Yoann Dabrowski.

15 recordsLinked to original sources

arXiv2017

Models of Linear Logic based on the Schwartz $\varepsilon$-product

From the interpretation of Linear Logic multiplicative disjunction as the $\varepsilon$-product defined by Laurent Schwartz, we construct several models of Differential Linear Logic based on usual mathematical notions of smooth maps. This improves on previous results, by R. Blute, T. Ehrhard and C. Tasson, based on convenient smoothness where only intuitionist models were built. We isolate a completeness condition, called k-quasi-completeness, and an associated notion stable by duality called k-reflexivity, allowing for a $*$-autonomous category of k-reflexive spaces in which the dual of the tensor product is the reflexive version of the $\varepsilon$ product. We adapt Meise's definition of Smooth maps into a first model of Differential Linear Logic, made of k-reflexive spaces. We also build two new models of Linear Logic with conveniently smooth maps, on categories made respectively of Mackey-complete Schwartz spaces and Mackey-complete Nuclear Spaces (with extra reflexivity conditions). Varying slightly the notion of smoothness, one also recovers models of DiLL on the same $*$-autonomous categories. Throughout the article, we work within the setting of Dialogue categories where the tensor product is exactly the $\varepsilon$-product (without reflexivization).

cs.LO↗

arXiv2017

A Laplace Principle for Hermitian Brownian Motion and Free Entropy I: the convex functional case

This paper is part of a series aiming at proving that the $\limsup$ and $\liminf$ variants of Voiculescu's free entropy coincide. This is based on a Laplace principle (implying a large deviation principle) for hermitian brownian motion on $[0,1]$. In the current paper, we show that microstates free entropy $χ(X_1,...,X_m)$ and non-microstate free entropy $χ^*(X_1,...,X_m)$ coincide for self-adjoint variables $(X_1,...,X_m)$ satisfying a Schwinger-Dyson equation for subquadratic, bounded below, strictly convex potentials with Lipschitz derivative sufficiently approximable by non-commutative polynomials. Our results are based on Dupuis-Ellis weak convergence approach to large deviations where one shows a Laplace principle in obtaining a stochastic control formulation for exponential functionals. In the non-commutative context, ultrapoduct analysis replaces weak-convergence of the stochastic control problems.

math.PR↗

arXiv2016

Linear rigidity of stationary stochastic processes

We consider stationary stochastic processes $X_n$, $n\in \mathbb{Z}$ such that $X_0$ lies in the closed linear span of $X_n$, $n\neq 0$; following Ghosh and Peres, we call such processes linearly rigid. Using a criterion of Kolmogorov, we show that it suffices, for a stationary stochastic process to be rigid, that the spectral density vanish at zero and belong to the Zygmund class $Λ_{*}(1)$. We next give sufficient condition for stationary determinantal point processes on $\mathbb{Z}$ and on $\mathbb{R}$ to be rigid. Finally, we show that the determinantal point process on $\mathbb{R}^2$ induced by a tensor square of Dyson sine-kernels is $\textit{not}$ linearly rigid.

math.PR↗

arXiv2016

Continuous model theories for von Neumann algebras

We axiomatize in (first order finitary) continuous logic for metric structures $σ$-finite $W^*$-probability spaces and preduals of von Neumann algebras jointly with a weak-* dense $C^*$-algebra of its dual. This corresponds to the Ocneanu ultrapower and the Groh ultrapower of ($σ$-finite in the first case) von Neumann algebras. We give various axiomatizability results corresponding to recent results of Ando and Haagerup including axiomatizability of $III_λ$ factors for $0<λ\leq 1$ fixed and their preduals. We also strengthen the concrete Groh theory to an axiomatization result for preduals of von Neumann algebras in the language of tracial matrix-ordered operator spaces, a natural language for preduals of dual operator systems. We give an application to the isomorphism of ultrapowers of factors of type $III$ and $II_\infty$ for different ultrafilters.

math.OA↗

arXiv2015

Analytic functions relative to a covariance map $η$: I. Generalized Haagerup products and analytic relations

We generalize module weak-* Haagerup tensor products to obtain complete quotients of normal Haagerup tensor product included in canonical Hilbert spaces associated to completely positive normal (covariance) maps $η$ on a finite von Neumann algebra $B$. We construct in this way dual operator spaces, providing new examples even in the case of module extended Haagerup tensor products. This is the basis for defining a matrix normed algebra of analytic functions that captures the relations of free semicircular variables with covariance $η$. We prove that a class of non-commutative random variables having finite Fisher information relative to $η$ have also no analytic relations among our class of analytic functions.

math.OA↗

arXiv2014

Functional properties of Generalized Hörmander spaces of distributions II : Multilinear maps and applications to spaces of functionals with wave front set conditions

We continue our study and applications of generalized Hörmander spaces of distributions $\mathcal{D}'_{γ,Λ}$ with $C^\infty$ wavefront set included in a cone $Λ$ and the union of $H^s$-wave front sets in a second cone $γ\subset Λ$. We give hypocontinuity results and failure of continuity of tensor multiplication maps between these spaces and deduce hypocontinuity results for various compositions on spaces of multilinear maps. We apply this study to a generalization of microcausal functionals from algebraic quantum field theory with derivatives controlled by spaces either of the form $\mathcal{D}'_{γ,Λ}$ or some $ε$-tensor product of them. We prove nuclearity and completeness results and give general results to build Poisson algebra structures (with at least hypocontinuous bilinear products). We also apply our general framework to build retarded products with field dependent propagators.

math-ph↗

arXiv2014

Functional properties of Generalized Hörmander spaces of distributions I : Duality theory, completions and bornologifications

The space $D'_Λ$ of distributions having their $C^\infty$ wavefront set in a cone $Λ$ has become important in physics because of its role in the formulation of quantum field theory in curved spacetime. It is also a basic object in microlocal analysis, but not well studied from a functional analytic viewpoint. In order to compute its completion in the open cone case, we introduce generalized spaces $D'_{γ,Λ}$ where we also control the union of H^s-wave front sets in a second cone $γ$ contained in $Λ$. We can compute bornological and topological duals, completions and bornologifications of natural topologies for spaces in this class. All our topologies are nuclear, ultrabornological when bornological and we can describe when they are quasi-LB. We also give concrete microlocal representations of bounded and equicontinuous sets in those spaces and work with general support conditions including future compact or space compact support conditions on globally hyperbolic manifolds, as motivated by physics applications to be developed in a second paper.

math.FA↗

arXiv2014

The simplex of tracial quantum symmetric states

We show that the space of tracial quantum symmetric states of an arbitrary unital C*-algebra is a Choquet simplex and is a face of the tracial state space of the universal unital C*-algebra free product of A with itself infinitely many times. We also show that the extreme points of this simplex are dense, making it the Poulsen simplex when A is separable and nontrivial. In the course of the proof we characterize the centers of certain tracial amalgamated free product C*-algebras.

math.OA↗

arXiv2014

Functional properties of Hörmander's space of distributions having a specified wavefront set

The space $D'_Γ$ of distributions having their wavefront sets in a closed cone $Γ$ has become important in physics because of its role in the formulation of quantum field theory in curved space time. In this paper, the topological and bornological properties of $D'_Γ$ and its dual $E'_Λ$ are investigated. It is found that $D'_Γ$ is a nuclear, semi-reflexive and semi-Montel complete normal space of distributions. Its strong dual $E'_Λ$ is a nuclear, barrelled and bornological normal space of distributions which, however, is not even sequentially complete. Concrete rules are given to determine whether a distribution belongs to $D'_Γ$, whether a sequence converges in $D'_Γ$ and whether a set of distributions is bounded in $D'_Γ$.

math-ph↗

arXiv2014

Time Reversal of free diffusions I : Reversed Brownian motion, Reversed SDE and first order regularity of conjugate variables

We show that solutions of free stochastic differential equations with regular drifts and diffusion coefficients, when considered backwards in time, still satisfy free SDEs for an explicit free Brownian motion and drift. We also study the stochastic integral part with respect to this reversed free Brownian motion of canonical closed martingales. We deduce that conjugate variables computed along a free Brownian motion, an example of such a reversed martingale appearing in the definition of non-microstates free entropy, are in the $L^2$ domain of corresponding free difference quotients for almost every time.

math.PR↗

arXiv2013

A Free Stochastic Partial Differential Equation

We get stationary solutions of a free stochastic partial differential equation. As an application, we prove equality of non-microstate and microstate free entropy dimensions under a Lipschitz like condition on conjugate variables, assuming also R^ω embeddability. This includes an N-tuple of q-Gaussian random variables e.g. for |q|N\leq 0.13.

math.OA↗

arXiv2012

Unbounded derivations, free dilations and indecomposability results for II$_1$ factors

We give sufficient conditions, in terms of the existence of unbounded derivations satisfying certain properties, which ensure that a II$_1$ factor $M$ is prime or has at most one Cartan subalgebra. For instance, we prove that if there exists a real closable unbounded densely defined derivation $δ:M\rightarrow L^2(M)\bar{\otimes}L^2(M)$ whose domain contains a non-amenability set, then $M$ is prime. If $δ$ is moreover "algebraic" (i.e. its domain $M_0$ is finitely generated, $δ(M_0)\subset M_0\otimes M_0$ and $δ^*(1\otimes 1)\in M_0$), then we show that $M$ has no Cartan subalgebra. We also give several applications to examples from free probability. Finally, we provide a class of countable groups $Γ$, defined through the existence of an unbounded cocycle $b:Γ\rightarrow \mathbb C(Γ/Λ)$, for some subgroup $Λ<Γ$, such that the II$_1$ factor $L^{\infty}(X)\rtimesΓ$ has a unique Cartan subalgebra, up to unitary conjugacy, for any free ergodic probability measure preserving (pmp) action $Γ\curvearrowright (X,μ)$.

math.OA↗

arXiv2012

Concavification of free entropy

We introduce a modification of Voiculescu's free entropy which coincides with the liminf variant of Voiculescu's free entropy on extremal states, but is a concave upper semi-continuous function on the trace state space. We also extend the orbital free entropy of Hiai, Miyamoto and Ueda to non-hyperfinite multivariables and prove freeness in case of additivity of Voiculescu's entropy (or vanishing of our extended orbital entropy).

math.OA↗

arXiv2010

A non-commutative Path Space approach to stationary free Stochastic Differential Equations

By defining tracial states on a non-commutative analogue of a path space, we construct Markov dilations for a class of conservative completely Markov semigroups on finite von Neumann algebras. This class includes all symmetric semigroups. For well chosen semigroups (for instance with generator any divergence form operator associated to a derivation valued in the coarse correspondence) those dilations give rise to stationary solutions of certain free SDEs previously considered by D. Shlyakhtenko. Among applications, we prove a non-commutative Talagrand inequality for non-microstates free entropy (relative to a subalgebra B and a completely positive map η:B\to B). We also use those new deformations in conjunction with Popa's deformation/rigidity techniques. For instance, combining our results with techniques of Popa-Ozawa and Peterson, we prove that the von Neumann algebra of a countable discrete group with CMAP and positive first L^2 Betti number has no Cartan subalgebras.

math.OA↗

arXiv2009

A Note about proving non-$Γ$ under a finite non-microstates free Fisher information Assumption

We prove that if $X_{1},...,X_{n} (n >1)$ are selfadjoints in a $W^{*}$-probability space with finite non-microstates free Fisher information, then the von Neumann algebra $W^{*}(X_{1},...,X_{n})$ they generate doesn't have property $Γ$ (especially is not amenable). This is an analog of a well-known result of Voiculescu for microstates free entropy. We also prove factoriality under finite non-microstates entropy.

math.OA↗