Triple operator version of the Golden-Thompson inequality for traces on von Neumann algebras

Randrianantoanina, Narcisse

doi:10.5802/aif.3575

Triple operator version of the Golden-Thompson inequality for traces on von Neumann algebras
[Une version triple operateur de l’inégalité Golden-Thompson pour les traces sur les algèbres de von Neumann]

Randrianantoanina, Narcisse ¹

¹ Department of Mathematics Miami University Oxford, Ohio 45056 (USA)

Annales de l'Institut Fourier, Online first, 41 p.

Résumé
Abstract

Nous prouvons une géneralisation de l’extension de Lieb à trois matrices de l’inégalité de Golden–Thompson de l’algèbre des matrices à des traces associés à des algèbres de von Neumann finies. Plus précisément, supposons que $ℳ$ est une algèbre de von Neumann finie munie d’un état tracé $τ$ . Soient $1 \leq p, q \leq \infty$ tels que $1 / p + 1 / q = 1$ . Alors pour tout $a$ , $b$ , et $c$ opérateurs auto-adjoints et $τ$ -mesurables satisfaisant $a \in ℳ$ , $e^{b} \in L_{p} (ℳ, τ)$ , et $e^{c} \in L_{q} (ℳ, τ)$ , on a l’inégalité suivante :

τ (e^{a + b + c}) \leq \int_{0}^{\infty} τ (e^{c / 2} {(e^{- a} + t 1)}^{- 1} e^{b} {(e^{- a} + t 1)}^{- 1} e^{c / 2}) d t

où $1$ denote l’identité de $ℳ$ . Nous présentons également d’autres résultats liés à Wigner–Yanase–Dyson–Lieb concavité dans le contexte général d’état tracé.

Nous utilisons la version ci-dessus de l’inégalité de Golden–Thompson pour trois opérateurs pour demontrer une extension de inégalité arcsinh de Prokhorov aux martingales non commutatives dans des espaces de probabilité non commutatif.

We provide a generalization of Lieb’s triple matrix extension of the Golden–Thompson inequality from matrix algebras to the setting of traces on finite von Neumann algebras. More precisely, assume that $ℳ$ is a finite von Neumann algebra equipped with a tracial state $τ$ . If $1 \leq p, q \leq \infty$ with $1 / p + 1 / q = 1$ , it is shown that whenever $a$ , $b$ , and $c$ are self-adjoint $τ$ -measurable operators satisfying $a \in ℳ$ , $e^{b} \in L_{p} (ℳ, τ)$ , and $e^{c} \in L_{q} (ℳ, τ)$ , then the following inequality holds:

τ (e^{a + b + c}) \leq \int_{0}^{\infty} τ (e^{c / 2} {(e^{- a} + t 1)}^{- 1} e^{b} {(e^{- a} + t 1)}^{- 1} e^{c / 2}) d t

where $1$ denotes the identity in $ℳ$ . We also present other results related to the Wigner–Yanase–Dyson–Lieb concavity in the context of general tracial state.

We use the above version of the Golden–Thompson inequality for three operators to prove an extension of the Prokhorov arcsinh inequality to noncommutative martingales in general noncommutative probability spaces.

Reçu le : 2021-09-27
Révisé le : 2022-05-20
Accepté le : 2022-09-23
Première publication : 2023-05-26

DOI : 10.5802/aif.3575

Classification : 46L51, 46L53, 15A16, 47A60, 15A15
Keywords: Trace inequalities, von Neumann algebras, noncommutative martingales
Mot clés : inégalités de trace, algèbres de von Neumann, martingales non commutatives

Affiliations des auteurs :

Randrianantoanina, Narcisse ¹

¹ Department of Mathematics Miami University Oxford, Ohio 45056 (USA)

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Randrianantoanina, Narcisse. Triple operator version of the Golden-Thompson inequality for traces on von Neumann algebras. Annales de l'Institut Fourier, Online first, 41 p.

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