Given a completely bounded map from an operator space into a von Neumann algebra (or merely a unital dual algebra) , we define to be -semidiscrete if for any operator algebra , the tensor operator is bounded from into , with norm less than . 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 -nuclearity to operators. Having in mind the equivalence “ is nuclear semidiscrete”, when is a -algebra, we then study the relationships between the nuclearity of an operator and the semidiscreteness of its decomposable norm on operators between -algebras. Lastly, we apply some of our techniques to find new properties of Haagerup’s decomposable norm on operators between -algebras.
Étant donnée une application complètement bornée d’un espace d’opérateurs dans une algèbre de von Neumann (ou simplement une algèbre duale unifère) , on dit que est -semi-discrète si pour toute algèbre d’opérateurs , le produit tensoriel d’opérateurs est borné de dans , avec une norme inférieure ou égale à . 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 -nucléarité pour les applications linéaires. Ayant à l’esprit l’équivalence “ est nucléaire est semi-discrète” valable pour une -algèbre , 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 -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}, pages = {1869--1901}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {49}, number = {6}, year = {1999}, doi = {10.5802/aif.1741}, zbl = {0989.46033}, mrnumber = {2001b:46092}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1741/} }
TY - JOUR AU - Merdy, Christian Le TI - Finite rank approximation and semidiscreteness for linear operators JO - Annales de l'Institut Fourier PY - 1999 SP - 1869 EP - 1901 VL - 49 IS - 6 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1741/ DO - 10.5802/aif.1741 LA - en ID - AIF_1999__49_6_1869_0 ER -
%0 Journal Article %A Merdy, Christian Le %T Finite rank approximation and semidiscreteness for linear operators %J Annales de l'Institut Fourier %D 1999 %P 1869-1901 %V 49 %N 6 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.1741/ %R 10.5802/aif.1741 %G en %F 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. https://aif.centre-mersenne.org/articles/10.5802/aif.1741/
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