A $C^*$-algebraic Schoenberg theorem

Bratteli, Ola ; Jorgensen, Palle E. T. ; Kishimoto, Akitaka ; Robinson, Donald W.

Annales de l'Institut Fourier, Tome 34 (1984), p. 155-187 / Harvested from Numdam

Résumé

Soient ${\frak A}$ une $C^*$-algèbre, $G$ un groupe compact abélien, $\tau $ une action de $G$ sur ${\frak A},{\frak A}^{\tau }$ la sous-algèbre des points fixes de $\tau $ et ${\frak A}_F$ la sous-algèbre dense de ${\frak A}$, des éléments $G$-finis. Soit ensuite $H$ un opérateur linéaire de ${\frak A}_ F$ dans ${\frak A}$ qui commute avec $\tau $ et qui est nul sur ${\frak A}^{\tau }$. Nous prouvons que $H$ est une dissipation complète si et seulement si $H$ est fermable et sa clôture est le générateur d’un semi-groupe de type $C_0$ de contractions complètement positives. Ces dissipations complètes sont classifiées à l’aide de certaines applications de type négatif tordu du groupe dual $\hat{G}$ dans des opérateurs dissipatifs, affiliés au centre de l’algèbre des multiplicateurs de ${\frak A}^{\tau }$. Dans ce cadre, il est également établi que les dissipations complètes forment un sous-ensemble propre des dissipations générales, sauf pour le cas où ${\frak A}$ est une $C^*$-algèbre abélienne.

Abstract

Let ${\frak A}$ be a $C^*$-algebra, $G$ a compact abelian group, $\tau $ an action of $G$ by $*$-automorphisms of ${\frak A},{\frak A}^{\tau }$ the fixed point algebra of $\tau $ and ${\frak A}_F$ the dense sub-algebra of $G$-finite elements in ${\frak A}$. Further let $H$ be a linear operator from ${\frak A}_ F$ into ${\frak A}$ which commutes with $\tau $ and vanishes on ${\frak A}^{\tau }$. We prove that $H$ is a complete dissipation if and only if $H$ is closable and its closure generates a $C_0$-semigroup of completely positive contractions. These complete dissipations are classified in terms of certain twisted negative definite maps from the dual group $\hat{G}$ into dissipative operators affiliated with the center of the multiplier algebra of ${\frak A}^{\tau }$. We also argue that the complete dissipation property is strictly stronger than the usual dissipation property, except in special circumstances such as when ${\frak A}$ is abelian.

Détail

Publié le : 1984-01-01

MR 86b:46105 | Zbl 0536.46046

DOI : https://doi.org/10.5802/aif.981

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