Finite rank approximation and semidiscreteness for linear operators

Merdy, Christian Le

Annales de l'Institut Fourier, Tome 49 (1999) no. 6, pp. 1869-1901.

Étant donnée une application complètement bornée $u:Z\to M$ 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 $I_A\otimes u$ est borné de $A{\otimes }_{\min }Z$ dans $A{\otimes }_{\mathrm{nor}}M$, 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 $\iff B^{**}$ 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.

Given a completely bounded map $u:Z\to M$ 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 $I_A\otimes u$ is bounded from $A{\otimes }_{\min }Z$ into $A{\otimes }_{\mathrm{nor}}M$, 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 $\iff B^{**}$ 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.

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