Finite rank approximation and semidiscreteness for linear operators
Annales de l'Institut Fourier, Volume 49 (1999) no. 6, p. 1869-1901

Given a completely bounded map u:ZM from an operator space Z into a von Neumann algebra (or merely a unital dual algebra) M, we define u to be C-semidiscrete if for any operator algebra A, the tensor operator I A u is bounded from A min Z into A nor M, with norm less than C. We investigate this property and characterize it by suitable approximation properties, thus generalizing the Choi-Effros characterization of semidiscrete von Neumann algebras. Our work is an extension of some recent work of Pisier on an analogous generalization of C * -nuclearity to operators. Having in mind the equivalence “B is nuclear B ** semidiscrete”, when B is a C * -algebra, we then study the relationships between the nuclearity of an operator and the semidiscreteness of its decomposable norm on operators between C * -algebras. Lastly, we apply some of our techniques to find new properties of Haagerup’s decomposable norm on operators between C * -algebras.

Étant donnée une application complètement bornée u:ZM d’un espace d’opérateurs Z dans une algèbre de von Neumann (ou simplement une algèbre duale unifère) M, on dit que u est C-semi-discrète si pour toute algèbre d’opérateurs A, le produit tensoriel d’opérateurs I A u est borné de A min Z dans A nor M, avec une norme inférieure ou égale à C. Nous étudions cette propriété et la caractérisons notamment par des propriétés d’approximation appropriées, qui généralisent la caractérisation des algèbres de von Neumann semi-discrètes due à Choi-Effros. Notre travail est une extension de travaux récents de Pisier sur une notion comparable de C * -nucléarité pour les applications linéaires. Ayant à l’esprit l’équivalence “B est nucléaire B ** est semi-discrète” valable pour une C * -algèbre B, nous étudions les relations qui existent entre la nucléarité d’une application linéaire et le caractère semi-discret de son biadjoint. Enfin nous obtenons, grâce à certaines de nos techniques, de nouvelles propriétés de la norme de Haagerup pour les opérateurs décomposables entre C * -algèbres.

@article{AIF_1999__49_6_1869_0,
     author = {Merdy, Christian Le},
     title = {Finite rank approximation and semidiscreteness for linear operators},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {49},
     number = {6},
     year = {1999},
     pages = {1869-1901},
     doi = {10.5802/aif.1741},
     zbl = {0989.46033},
     mrnumber = {2001b:46092},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1999__49_6_1869_0}
}
Merdy, Christian Le. Finite rank approximation and semidiscreteness for linear operators. Annales de l'Institut Fourier, Volume 49 (1999) no. 6, pp. 1869-1901. doi : 10.5802/aif.1741. http://www.numdam.org/item/AIF_1999__49_6_1869_0/

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