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 . If with , it is shown that whenever , , and are self-adjoint -measurable operators satisfying , , and , then the following inequality holds:
where 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 . Soient tels que . Alors pour tout , , et opérateurs auto-adjoints et -mesurables satisfaisant , , et , on a l’inégalité suivante :
où 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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Keywords: Trace inequalities, von Neumann algebras, noncommutative martingales
Mot clés : inégalités de trace, algèbres de von Neumann, martingales non commutatives
Randrianantoanina, Narcisse 1
@article{AIF_2024__74_1_193_0, author = {Randrianantoanina, Narcisse}, title = {Triple operator version of the {Golden-Thompson} inequality for traces on von {Neumann} algebras}, journal = {Annales de l'Institut Fourier}, pages = {193--233}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {74}, number = {1}, year = {2024}, doi = {10.5802/aif.3575}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3575/} }
TY - JOUR AU - Randrianantoanina, Narcisse TI - Triple operator version of the Golden-Thompson inequality for traces on von Neumann algebras JO - Annales de l'Institut Fourier PY - 2024 SP - 193 EP - 233 VL - 74 IS - 1 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3575/ DO - 10.5802/aif.3575 LA - en ID - AIF_2024__74_1_193_0 ER -
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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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