For a normalized positive definite function on a locally compact abelian group , let be the unitary representation associated to by the GNS construction. We give necessary and sufficient conditions for the vanishing of 1-cohomology and reduced 1-cohomology . For example, if and only if either or , where is the trivial character of and is the probability measure on the Pontryagin dual associated to by Bochner’s Theorem. This streamlines an argument of Guichardet (see Theorem 4 in [7]).
Soit une fonction de type positif normalisée sur un groupe localement compact abélien , et la représentation unitaire de obtenue par construction GNS. Nous donnons des conditions nécessaires et suffisantes pour l’annulation de la 1-cohomologie et de la 1-cohomologie réduite . Par exemple, si et seulement si ou bien ou bien , où est le caractère trivial de et est la mesure de probabilité sur le dual de Pontryagin associée à par le théorème de Bochner. Cela simplifie un argument de Guichardet (Théorème 4 de [7]).
@article{AMBP_2014__21_1_61_0, author = {Jordan Franks and Alain Valette}, title = {On $1$-cocycles induced by a positive definite function on a locally compact abelian group}, journal = {Annales math\'ematiques Blaise Pascal}, pages = {61--69}, publisher = {Annales math\'ematiques Blaise Pascal}, volume = {21}, number = {1}, year = {2014}, doi = {10.5802/ambp.335}, mrnumber = {3248221}, zbl = {1297.43007}, language = {en}, url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.335/} }
TY - JOUR AU - Jordan Franks AU - Alain Valette TI - On $1$-cocycles induced by a positive definite function on a locally compact abelian group JO - Annales mathématiques Blaise Pascal PY - 2014 SP - 61 EP - 69 VL - 21 IS - 1 PB - Annales mathématiques Blaise Pascal UR - https://ambp.centre-mersenne.org/articles/10.5802/ambp.335/ DO - 10.5802/ambp.335 LA - en ID - AMBP_2014__21_1_61_0 ER -
%0 Journal Article %A Jordan Franks %A Alain Valette %T On $1$-cocycles induced by a positive definite function on a locally compact abelian group %J Annales mathématiques Blaise Pascal %D 2014 %P 61-69 %V 21 %N 1 %I Annales mathématiques Blaise Pascal %U https://ambp.centre-mersenne.org/articles/10.5802/ambp.335/ %R 10.5802/ambp.335 %G en %F AMBP_2014__21_1_61_0
Jordan Franks; Alain Valette. On $1$-cocycles induced by a positive definite function on a locally compact abelian group. Annales mathématiques Blaise Pascal, Volume 21 (2014) no. 1, pp. 61-69. doi : 10.5802/ambp.335. https://ambp.centre-mersenne.org/articles/10.5802/ambp.335/
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