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, Tome 74 (2024) no. 1, pp. 193-233.

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
Publié le : 2024-05-17

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)

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@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  -

%0 Journal Article
%A Randrianantoanina, Narcisse
%T Triple operator version of the Golden-Thompson inequality for traces on von Neumann algebras
%J Annales de l'Institut Fourier
%D 2024
%P 193-233
%V 74
%N 1
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3575/
%R 10.5802/aif.3575
%G en
%F AIF_2024__74_1_193_0

Randrianantoanina, Narcisse. Triple operator version of the Golden-Thompson inequality for traces on von Neumann algebras. Annales de l'Institut Fourier, Tome 74 (2024) no. 1, pp. 193-233. doi : 10.5802/aif.3575. https://aif.centre-mersenne.org/articles/10.5802/aif.3575/

Bibliographie
Cité par

[1] Araki, Huzihiro Golden-Thompson and Peierls-Bogolubov inequalities for a general von Neumann algebra, Commun. Math. Phys., Volume 34 (1973), pp. 167-178 | DOI | MR | Zbl

[2] Araki, Huzihiro Relative entropy of states of von Neumann algebras, Publ. Res. Inst. Math. Sci., Volume 11 (1976) no. 3, pp. 809-833 | DOI | MR | Zbl

[3] Bhatia, Rajendra Matrix analysis, Graduate Texts in Mathematics, 169, Springer, 1996, xii+347 pages | DOI

[4] Chen, Zeqian; Randrianantoanina, Narcisse; Xu, Quanhua Atomic decompositions for noncommutative martingales (2020) (https://arxiv.org/abs/2001.08775v1)

[5] Diestel, Joseph; Uhl, Jerry J. Jr Vector measures, Math. Surveys, 15, American Mathematical Society, 1977, xiii+322 pages | DOI

[6] Dodds, Peter G.; Dodds, Theresa K.-Y.; de Pagter, Ben Noncommutative Köthe duality, Trans. Am. Math. Soc., Volume 339 (1993) no. 2, pp. 717-750 | MR | Zbl

[7] Fack, Thierry; Kosaki, Hideki Generalized $s$ -numbers of $τ$ -measurable operators, Pac. J. Math., Volume 123 (1986), pp. 269-300 | DOI | MR | Zbl

[8] Forrester, Peter J.; Thompson, Colin J. The Golden-Thompson inequality: historical aspects and random matrix applications, J. Math. Phys., Volume 55 (2014) no. 2, 023503, 12 pages | DOI | MR | Zbl

[9] Golden, Sidney Lower bounds for the Helmholtz function, Phys. Rev., II. Ser., Volume 137 (1965), p. B1127-B1128 | MR | Zbl

[10] Groh, Ulrich Uniform ergodic theorems for identity preserving Schwarz maps on $W^{*}$ -algebras, J. Oper. Theory, Volume 11 (1984) no. 2, pp. 395-404 | MR | Zbl

[11] Hansen, Frank; Pedersen, Gert K. Perturbation formulas for traces on $C^{*}$ -algebras, Publ. Res. Inst. Math. Sci., Volume 31 (1995) no. 1, pp. 169-178 | DOI | MR | Zbl

[12] Heinrich, Stefan Ultraproducts in Banach space theory, J. Reine Angew. Math., Volume 313 (1980), pp. 72-104 | DOI | MR | Zbl

[13] Hitczenko, Paweł Best constants in martingale version of Rosenthal’s inequality, Ann. Probab., Volume 18 (1990) no. 4, pp. 1656-1668 | MR | Zbl

[14] Jiao, Yong; Osękowski, Adam; Wu, Lian Inequalities for noncommutative differentially subordinate martingales, Adv. Math., Volume 337 (2018) no. 1, pp. 216-259 | DOI | MR | Zbl

[15] Johnson, William B.; Schechtman, Gideon; Zinn, Joel Best constants in moment inequalities for linear combinations of independent and exchangeable random variables, Ann. Probab., Volume 13 (1985) no. 1, pp. 234-253 | MR | Zbl

[16] Junge, Marius Doob’s inequality for non-commutative martingales, J. Reine Angew. Math., Volume 549 (2002), pp. 149-190 | MR | Zbl

[17] Junge, Marius; Perrin, Mathilde Theory of $ℋ_{p}$ -spaces for continuous filtrations in von Neumann algebras, Astérisque, 362, Société Mathématique de France, 2014, vi+134 pages | Numdam

[18] Junge, Marius; Xu, Quanhua Noncommutative Burkholder/Rosenthal inequalities, Ann. Probab., Volume 31 (2003) no. 2, pp. 948-995 | MR | Zbl

[19] Junge, Marius; Xu, Quanhua Noncommutative Burkholder/Rosenthal inequalities. II. Applications, Isr. J. Math., Volume 167 (2008), pp. 227-282 | DOI | MR | Zbl

[20] Junge, Marius; Zeng, Qiang Noncommutative Bennett and Rosenthal inequalities, Ann. Probab., Volume 41 (2013) no. 6, pp. 4287-4316 | DOI | MR | Zbl

[21] Junge, Marius; Zeng, Qiang Noncommutative martingale deviation and Poincaré type inequalities with applications, Probab. Theory Relat. Fields, Volume 161 (2015) no. 3-4, pp. 449-507 | DOI | Zbl

[22] Kalton, Nigel J.; Sukochev, Fedor Symmetric norms and spaces of operators, J. Reine Angew. Math., Volume 621 (2008), pp. 81-121 | DOI | MR | Zbl

[23] Kosaki, Hideki Interpolation theory and the Wigner–Yanase–Dyson–Lieb concavity, Commun. Math. Phys., Volume 87 (1982) no. 3, pp. 315-329 | DOI | MR | Zbl

[24] Kosaki, Hideki An inequality of Araki–Lieb–Thirring (von Neumann algebra case), Proc. Am. Math. Soc., Volume 114 (1992) no. 2, pp. 477-481 | DOI | MR | Zbl

[25] Lieb, Elliott Convex trace functions and the Wigner–Yanase–Dyson conjecture, Adv. Math., Volume 11 (1973), pp. 267-288 | DOI | MR | Zbl

[26] Lieb, Elliott; Ruskai, Mary B. A fundamental property of quantum-mechanical entropy, Phys. Rev. Lett., Volume 30 (1973), pp. 434-436 | DOI | MR

[27] McDuff, Dusa Uncountably many ${II}_{1}$ factors, Ann. Math., Volume 90 (1969), pp. 372-377 | DOI | MR

[28] Nelson, Edward Notes on non-commutative integration, J. Funct. Anal., Volume 15 (1974), pp. 103-116 | DOI | MR | Zbl

[29] Ocneanu, Adrian Actions of discrete amenable groups on von Neumann algebras, Lecture Notes in Mathematics, 1138, Springer, 1985, iv+115 pages | DOI | MR

[30] Petz, Dénes A survey of certain trace inequalities, Functional analysis and operator theory (Warsaw, 1992) (Banach Center Publications), Volume 30, Polish Academy of Sciences, 1994, pp. 287-298 | MR | Zbl

[31] Pisier, Gilles; Xu, Quanhua Non-commutative martingale inequalities, Commun. Math. Phys., Volume 189 (1997) no. 3, pp. 667-698 | DOI | MR | Zbl

[32] Pisier, Gilles; Xu, Quanhua Non-commutative $L^{p}$ -spaces, Handbook of the geometry of Banach spaces, Vol. 2, North-Holland, 2003, pp. 1459-1517 | DOI | Zbl

[33] Prokhorov, Yuriĭ V. An extremal problem in probability theory, Theor. Probability Appl., Volume 4 (1960), pp. 201-203 | DOI | MR | Zbl

[34] Randrianantoanina, Narcisse Non-commutative martingale transforms, J. Funct. Anal., Volume 194 (2002), pp. 181-212 | MR | Zbl

[35] Randrianantoanina, Narcisse; Wu, Lian Noncommutative Burkholder/Rosenthal inequalities associated with convex functions, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 53 (2017) no. 4, pp. 1575-1605 | DOI | MR | Zbl

[36] Raynaud, Yves On ultrapowers of non commutative $L_{p}$ spaces, J. Oper. Theory, Volume 48 (2002) no. 1, pp. 41-68 | MR | Zbl

[37] Ricard, Éric; Xu, Quanhua A noncommutative martingale convexity inequality, Ann. Probab., Volume 44 (2016) no. 2, pp. 867-882 | DOI | MR | Zbl

[38] Ruskai, Mary B. Inequalities for traces on von Neumann algebras, Commun. Math. Phys., Volume 26 (1972), pp. 280-289 | DOI | MR | Zbl

[39] Sadeghi, Ghadir; Moslehian, Mohammad Sal Noncommutative martingale concentration inequalities, Ill. J. Math., Volume 58 (2014) no. 2, pp. 561-575 | MR | Zbl

[40] Simon, Barry Trace ideals and their applications, Mathematical Surveys and Monographs, 120, American Mathematical Society, 2005, viii+150 pages | DOI

[41] Sims, Brailey “Ultra”-techniques in Banach space theory, Queen’s Papers in Pure and Applied Mathematics, 60, Queen’s University, 1982, iv+117 pages

[42] Sukochev, Fedor Completeness of quasi-normed symmetric operator spaces, Indag. Math., New Ser., Volume 25 (2014) no. 2, pp. 376-388 | DOI | MR | Zbl

[43] Sukochev, Fedor; Zanin, Dmitriy Johnson–Schechtman inequalities in the free probability theory, J. Funct. Anal., Volume 263 (2012) no. 10, pp. 2921-2948 | DOI | MR | Zbl

[44] Sutter, David; Berta, Mario; Tomamichel, Marco Multivariate trace inequalities, Commun. Math. Phys., Volume 352 (2017) no. 1, pp. 37-58 | DOI | MR | Zbl

[45] Thompson, Colin J. Inequality with applications in statistical mechanics, J. Math. Phys., Volume 6 (1965), pp. 1812-1813 | DOI | MR

[46] Tikhonov, Oleg E. Continuity of operator functions in topologies connected with a trace on a von Neumann algebra, Izv. Vyssh. Uchebn. Zaved., Mat. (1987) no. 1, pp. 77-79 | MR

[47] Tropp, Joel A. User-friendly tail bounds for sums of random matrices, Found. Comput. Math., Volume 12 (2012) no. 4, pp. 389-434 | DOI | MR | Zbl

[48] Vesterstrøm, Jørgen Quotients of finite $W^{*}$ -algebras, J. Funct. Anal., Volume 9 (1972), pp. 322-335 | DOI | MR | Zbl

[49] Xu, Quanhua Analytic functions with values in lattices and symmetric spaces of measurable operators, Math. Proc. Camb. Philos. Soc., Volume 109 (1991), pp. 541-563 | MR | Zbl

[50] Zhang, Haonan From Wigner–Yanase–Dyson conjecture to Carlen–Frank–Lieb conjecture, Adv. Math., Volume 365 (2020), 107053, 18 pages | DOI | MR | Zbl

Cité par Sources :

ANNALES DE L'INSTITUT FOURIER

ARTICLES À PARAÎTRE

FEUILLETER

PRESENTATION

COMITÉ DE RÉDACTION

SOUMETTRE UN ARTICLE

ABONNEMENT