A $C^{*}$-algebraic Schoenberg theorem

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

Annales de l'Institut Fourier, Tome 34 (1984) no. 3, pp. 155-187.

Soient $𝔄$ une $C^{*}$-algèbre, $G$ un groupe compact abélien, $\tau$ une action de $G$ sur $𝔄,{𝔄}^{\tau }$ la sous-algèbre des points fixes de $\tau$ et ${𝔄}_F$ la sous-algèbre dense de $𝔄$, des éléments $G$-finis. Soit ensuite $H$ un opérateur linéaire de ${𝔄}_F$ dans $𝔄$ qui commute avec $\tau$ et qui est nul sur ${𝔄}^{\tau }$. 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 $\widehat{G}$ dans des opérateurs dissipatifs, affiliés au centre de l’algèbre des multiplicateurs de ${𝔄}^{\tau }$. Dans ce cadre, il est également établi que les dissipations complètes forment un sous-ensemble propre des dissipations générales, sauf pour le cas où $𝔄$ est une $C^{*}$-algèbre abélienne.

Let $𝔄$ be a $C^{*}$-algebra, $G$ a compact abelian group, $\tau$ an action of $G$ by $*$-automorphisms of $𝔄,{𝔄}^{\tau }$ the fixed point algebra of $\tau$ and ${𝔄}_F$ the dense sub-algebra of $G$-finite elements in $𝔄$. Further let $H$ be a linear operator from ${𝔄}_F$ into $𝔄$ which commutes with $\tau$ and vanishes on ${𝔄}^{\tau }$. 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 $\widehat{G}$ into dissipative operators affiliated with the center of the multiplier algebra of ${𝔄}^{\tau }$. We also argue that the complete dissipation property is strictly stronger than the usual dissipation property, except in special circumstances such as when $𝔄$ is abelian.

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