Fréchet differentiability of the norm of $ {L}_{p}$-spaces associated with arbitrary von Neumann algebras

Denis Potapov; Fedor Sukochev; Anna Tomskova; Dmitriy Zanin

doi:10.1016/j.crma.2014.09.017

Functional analysis

Fréchet differentiability of the norm of

L_{p}

-spaces associated with arbitrary von Neumann algebras
[Différentiabilité au sens de Fréchet de la norme d'un espace

L_{p}

associé à une algèbre de von Neumann arbitraire]

Présenté par : Gilles Pisier

Denis Potapov ¹ ; Fedor Sukochev ¹ ; Anna Tomskova ¹ ; Dmitriy Zanin ¹

¹ School of Mathematics and Statistics, University of New South Wales, Kensington, NSW, 2052, Australia

Comptes Rendus. Mathématique, Volume 352 (2014) no. 11, pp. 923-927.

Résumé
Abstract

Soit $M$ une algèbre de von Neumann et soit $(L_{p} (M), {‖ \cdot ‖}_{p})$ , $1 \leq p < \infty$ l'espace $L_{p}$ de Haagerup sur $M$ . On montre que les propriétés de différentiabilité de ${‖ \cdot ‖}_{p}$ sont exactement les mêmes que celles obtenues sur les espaces $L_{p}$ classiques (commutatifs). Les ingrédients principaux sont les opérateurs intégraux multiples et les traces singulières.

Let $M$ be a von Neumann algebra and let $(L_{p} (M), {‖ \cdot ‖}_{p})$ , $1 \leq p < \infty$ be Haagerup's $L_{p}$ -space on $M$ . We prove that the differentiability properties of ${‖ \cdot ‖}_{p}$ are precisely the same as those of classical (commutative) $L_{p}$ -spaces. Our main instruments are multiple operator integrals and singular traces.

Reçu le : 2014-09-12
Accepté le : 2014-09-19
Publié le : 2014-10-01

DOI : 10.1016/j.crma.2014.09.017

Affiliations des auteurs :

Denis Potapov ¹ ; Fedor Sukochev ¹ ; Anna Tomskova ¹ ; Dmitriy Zanin ¹

¹ School of Mathematics and Statistics, University of New South Wales, Kensington, NSW, 2052, Australia

@article{CRMATH_2014__352_11_923_0,
     author = {Denis Potapov and Fedor Sukochev and Anna Tomskova and Dmitriy Zanin},
     title = {Fr\'echet differentiability of the norm of $ {L}_{p}$-spaces associated with arbitrary von {Neumann} algebras},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {923--927},
     publisher = {Elsevier},
     volume = {352},
     number = {11},
     year = {2014},
     doi = {10.1016/j.crma.2014.09.017},
     language = {en},
}

TY  - JOUR
AU  - Denis Potapov
AU  - Fedor Sukochev
AU  - Anna Tomskova
AU  - Dmitriy Zanin
TI  - Fréchet differentiability of the norm of $ {L}_{p}$-spaces associated with arbitrary von Neumann algebras
JO  - Comptes Rendus. Mathématique
PY  - 2014
SP  - 923
EP  - 927
VL  - 352
IS  - 11
PB  - Elsevier
DO  - 10.1016/j.crma.2014.09.017
LA  - en
ID  - CRMATH_2014__352_11_923_0
ER  -

%0 Journal Article
%A Denis Potapov
%A Fedor Sukochev
%A Anna Tomskova
%A Dmitriy Zanin
%T Fréchet differentiability of the norm of $ {L}_{p}$-spaces associated with arbitrary von Neumann algebras
%J Comptes Rendus. Mathématique
%D 2014
%P 923-927
%V 352
%N 11
%I Elsevier
%R 10.1016/j.crma.2014.09.017
%G en
%F CRMATH_2014__352_11_923_0

Denis Potapov; Fedor Sukochev; Anna Tomskova; Dmitriy Zanin. Fréchet differentiability of the norm of $ {L}_{p}$-spaces associated with arbitrary von Neumann algebras. Comptes Rendus. Mathématique, Volume 352 (2014) no. 11, pp. 923-927. doi : 10.1016/j.crma.2014.09.017. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.09.017/

Bibliographie
Cité par

[1] N.A. Azamov; A.L. Carey; P.G. Dodds; F.A. Sukochev Operator integrals, spectral shift, and spectral flow, Can. J. Math., Volume 61 (2009) no. 2, pp. 241-263

[2] M.S. Birman; M.Z. Solomyak Double Stieltjes operator integrals, Problems of Mathematical Physics, Izdat. Leningrad. Univ., Leningrad, 1966, pp. 33-67 (in Russian). English translation in: Topics in Mathematical Physics, vol. 1, Consultants Bureau Plenum Publishing Corporation, New York, 1967, pp. 25–54

[3] M.S. Birman; M.Z. Solomyak Double Stieltjes operator integrals II, Problems of Mathematical Physics, vol. 2, Izdat. Leningrad. Univ., Leningrad, 1967, pp. 26-60 (in Russian). English translation in: Topics in Mathematical Physics, vol. 2, Consultants Bureau, New York, 1968, pp. 19–46

[4] M.S. Birman; M.Z. Solomyak Double Stieltjes operator integrals III, Problems of Mathematical Physics, vol. 6, Leningrad University, Leningrad, 1973, pp. 27-53 (in Russian)

[5] R. Bonic; J. Frampton Smooth functions on Banach manifolds, J. Math. Mech., Volume 15 (1966) no. 5, pp. 877-898

[6] B. de Pagter; F.A. Sukochev Differentiation of operator functions in non-commutative $L_{p}$ -spaces, J. Funct. Anal., Volume 212 (2004) no. 1, pp. 28-75

[7] B. de Pagter; H. Witvliet; F.A. Sukochev Double operator integrals, J. Funct. Anal., Volume 192 (2002) no. 1, pp. 52-111

[8] T. Fack; H. Kosaki Generalized s-numbers of τ-measurable operators, Pac. J. Math., Volume 123 (1986) no. 2, pp. 269-300

[9] U. Haagerup $L_{p}$ -spaces associated with an arbitrary von Neumann algebra, Marseille (1977), pp. 175-184 (French summary: Algèbres d'opérateurs et leurs applications en physique mathématique)

[10] R.V. Kadison; J.R. Ringrose Fundamentals of the Theory of Operator Algebras, vol. II. Advanced Theory, Pure and Applied Mathematics, vol. 100, Academic Press, Inc., Orlando, FL, USA, 1986 (pp. ixiv and 399–1074)

[11] S. Lord; F. Sukochev; D. Zanin Singular Traces: Theory and Applications, Studies in Mathematics, vol. 46, De Gruyter, 2012

[12] L.A. Lusternik; V.L. Sobolev Elements of Functional Analysis, Frederick Ungar Publishing Co., New York, 1961 (Translated from Russian)

[13] V.V. Peller Multiple operator integrals and higher operator derivatives, J. Funct. Anal., Volume 233 (2006) no. 2, pp. 515-544

[14] G. Pisier; Q. Xu (Handbook of the Geometry of Banach Spaces), Volume vol. 2, North-Holland, Amsterdam (2003), pp. 1459-1517

[15] D. Potapov; F. Sukochev Fréchet differentiability of $S^{p}$ norms, Adv. Math., Volume 262 (2014), pp. 436-475

[16] D. Potapov; A. Skripka; F. Sukochev Spectral shift function of higher order, Invent. Math., Volume 193 (2013) no. 3, pp. 501-538

[17] K. Sundaresan Smooth Banach spaces, Math. Ann., Volume 173 (1967), pp. 191-199

[18] M. Terp $L_{p}$ spaces associated with von Neumann algebras, Math. Institute, Copenhagen University, 1981 (Notes)

[19] A. Thiago Bernardino A simple natural approach to the Uniform Boundedness Principle for multilinear mappings, Proyecciones, Volume 28 (2009) no. 3, pp. 203-207

[20] N. Tomczak-Jaegermann On the differentiability of the norm in trace classes $S_{p}$ , Séminaire Maurey–Schwartz 1974–1975: Espaces Lp$ {L}^{p}$, Applications radonifiantes et géométrie des espaces de Banach, Exp. No. XXII, Centre de mathématiques de l'École polytechnique, Paris, 1975

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Derivations with values in noncommutative symmetric spaces

Jinghao Huang; Fedor Sukochev

C. R. Math (2023)

Théorèmes ergodiques maximaux dans les espaces $L_{𝐩}$ non commutatifs

Marius Junge; Quanhua Xu

C. R. Math (2002)

Calcul fonctionnel et fonctions carrées dans les espaces L^p non commutatifs

Marius Junge; Christian Le Merdy; Quanhua Xu

C. R. Math (2003)