Let be a -algebra, a compact abelian group, an action of by -automorphisms of the fixed point algebra of and the dense sub-algebra of -finite elements in . Further let be a linear operator from into which commutes with and vanishes on . We prove that is a complete dissipation if and only if is closable and its closure generates a -semigroup of completely positive contractions. These complete dissipations are classified in terms of certain twisted negative definite maps from the dual group into dissipative operators affiliated with the center of the multiplier algebra of . We also argue that the complete dissipation property is strictly stronger than the usual dissipation property, except in special circumstances such as when is abelian.
Soient une -algèbre, un groupe compact abélien, une action de sur la sous-algèbre des points fixes de et la sous-algèbre dense de , des éléments -finis. Soit ensuite un opérateur linéaire de dans qui commute avec et qui est nul sur . Nous prouvons que est une dissipation complète si et seulement si est fermable et sa clôture est le générateur d’un semi-groupe de type de contractions complètement positives. Ces dissipations complètes sont classifiées à l’aide de certaines applications de type négatif tordu du groupe dual dans des opérateurs dissipatifs, affiliés au centre de l’algèbre des multiplicateurs de . Dans ce cadre, il est également établi que les dissipations complètes forment un sous-ensemble propre des dissipations générales, sauf pour le cas où est une -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}, pages = {155--187}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {34}, number = {3}, year = {1984}, doi = {10.5802/aif.981}, mrnumber = {86b:46105}, zbl = {0536.46046}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.981/} }
TY - JOUR AU - Bratteli, Ola AU - Jorgensen, Palle E. T. AU - Kishimoto, Akitaka AU - Robinson, Donald W. TI - A $C^*$-algebraic Schoenberg theorem JO - Annales de l'Institut Fourier PY - 1984 SP - 155 EP - 187 VL - 34 IS - 3 PB - Institut Fourier PP - Grenoble UR - http://archive.numdam.org/articles/10.5802/aif.981/ DO - 10.5802/aif.981 LA - en ID - AIF_1984__34_3_155_0 ER -
%0 Journal Article %A Bratteli, Ola %A Jorgensen, Palle E. T. %A Kishimoto, Akitaka %A Robinson, Donald W. %T A $C^*$-algebraic Schoenberg theorem %J Annales de l'Institut Fourier %D 1984 %P 155-187 %V 34 %N 3 %I Institut Fourier %C Grenoble %U http://archive.numdam.org/articles/10.5802/aif.981/ %R 10.5802/aif.981 %G en %F 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://archive.numdam.org/articles/10.5802/aif.981/
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