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

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

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

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

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

Comptes Rendus. Mathématique, Tome 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 :

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

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

@article{CRMATH_2014__352_11_923_0,
     author = {Potapov, Denis and Sukochev, Fedor and Tomskova, Anna and Zanin, Dmitriy},
     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},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2014.09.017/}
}

TY  - JOUR
AU  - Potapov, Denis
AU  - Sukochev, Fedor
AU  - Tomskova, Anna
AU  - Zanin, Dmitriy
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
UR  - http://www.numdam.org/articles/10.1016/j.crma.2014.09.017/
DO  - 10.1016/j.crma.2014.09.017
LA  - en
ID  - CRMATH_2014__352_11_923_0
ER  -

%0 Journal Article
%A Potapov, Denis
%A Sukochev, Fedor
%A Tomskova, Anna
%A Zanin, Dmitriy
%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
%U http://www.numdam.org/articles/10.1016/j.crma.2014.09.017/
%R 10.1016/j.crma.2014.09.017
%G en
%F CRMATH_2014__352_11_923_0

Potapov, Denis; Sukochev, Fedor; Tomskova, Anna; Zanin, Dmitriy. Fréchet differentiability of the norm of $ {L}_{p}$-spaces associated with arbitrary von Neumann algebras. Comptes Rendus. Mathématique, Tome 352 (2014) no. 11, pp. 923-927. doi : 10.1016/j.crma.2014.09.017. http://www.numdam.org/articles/10.1016/j.crma.2014.09.017/

Bibliographie
Cité par

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

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

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

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

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

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

[9] Haagerup, U. $L_{p}$ -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] Kadison, R.V.; Ringrose, J.R. 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] Lord, S.; Sukochev, F.; Zanin, D. Singular Traces: Theory and Applications, Studies in Mathematics, vol. 46, De Gruyter, 2012

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

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

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

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

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

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

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

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

[20] Tomczak-Jaegermann, N. 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 :