A C * -algebraic Schoenberg theorem
Annales de l'Institut Fourier, Volume 34 (1984) no. 3, p. 155-187

Let 𝔄 be a C * -algebra, G a compact abelian group, τ an action of G by *-automorphisms of 𝔄,𝔄 τ the fixed point algebra of τ 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 τ and vanishes on 𝔄 τ . 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 G ^ into dissipative operators affiliated with the center of the multiplier algebra of 𝔄 τ . 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.

Soient 𝔄 une C * -algèbre, G un groupe compact abélien, τ une action de G sur 𝔄,𝔄 τ la sous-algèbre des points fixes de τ 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 τ et qui est nul sur 𝔄 τ . 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 G ^ dans des opérateurs dissipatifs, affiliés au centre de l’algèbre des multiplicateurs de 𝔄 τ . 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.

@article{AIF_1984__34_3_155_0,
     author = {Bratteli, Ola and Jorgensen, Palle E. T. and Kishimoto, Akitaka and Robinson, Donald W.},
     title = {A $C^*$-algebraic Schoenberg theorem},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {34},
     number = {3},
     year = {1984},
     pages = {155-187},
     doi = {10.5802/aif.981},
     zbl = {0536.46046},
     mrnumber = {86b:46105},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1984__34_3_155_0}
}
Bratteli, Ola; Jorgensen, Palle E. T.; Kishimoto, Akitaka; Robinson, Donald W. A $C^*$-algebraic Schoenberg theorem. Annales de l'Institut Fourier, Volume 34 (1984) no. 3, pp. 155-187. doi : 10.5802/aif.981. http://www.numdam.org/item/AIF_1984__34_3_155_0/

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