Triple operator version of the Golden-Thompson inequality for traces on von Neumann algebras
Annales de l'Institut Fourier, Volume 74 (2024) no. 1, pp. 193-233.

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 is a finite von Neumann algebra equipped with a tracial state τ. If 1p,q with 1/p+1/q=1, it is shown that whenever a, b, and c are self-adjoint τ-measurable operators satisfying a, e b L p (,τ), and e c L q (,τ), then the following inequality holds:

τe a+b+c 0 τe c/2 (e -a +t1) -1 e b (e -a +t1) -1 e c/2 dt

where 1 denotes the identity in . 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.

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 est une algèbre de von Neumann finie munie d’un état tracé τ. Soient 1p,q tels que 1/p+1/q=1. Alors pour tout a, b, et c opérateurs auto-adjoints et τ-mesurables satisfaisant a, e b L p (,τ), et e c L q (,τ), on a l’inégalité suivante :

τe a+b+c 0 τe c/2 (e -a +t1) -1 e b (e -a +t1) -1 e c/2 dt

1 denote l’identité de . 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.

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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

Randrianantoanina, Narcisse 1

1 Department of Mathematics Miami University Oxford, Ohio 45056 (USA)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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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, Volume 74 (2024) no. 1, pp. 193-233. doi : 10.5802/aif.3575. https://aif.centre-mersenne.org/articles/10.5802/aif.3575/

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