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

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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. https://www.numdam.org/articles/10.1016/j.crma.2014.09.017/

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