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Comptes Rendus Mathématique
Volume 352, n° 11
pages 923-927 (novembre 2014)
Doi : 10.1016/j.crma.2014.09.017
Received : 12 September 2014 ;  accepted : 19 September 2014
Fréchet differentiability of the norm of  -spaces associated with arbitrary von Neumann algebras
Différentiabilité au sens de Fréchet de la norme d'un espace   associé à une algèbre de von Neumann arbitraire
 

Denis Potapov , Fedor Sukochev , Anna Tomskova , Dmitriy Zanin
 School of Mathematics and Statistics, University of New South Wales, Kensington, NSW, 2052, Australia 

Abstract

Let   be a von Neumann algebra and let  ,   be Haagerup's  -space on  . We prove that the differentiability properties of   are precisely the same as those of classical (commutative)  -spaces. Our main instruments are multiple operator integrals and singular traces.

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Résumé

Soit   une algèbre de von Neumann et soit  ,   l'espace   de Haagerup sur  . On montre que les propriétés de différentiabilité de   sont exactement les mêmes que celles obtenues sur les espaces   classiques (commutatifs). Les ingrédients principaux sont les opérateurs intégraux multiples et les traces singulières.

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