On 1-cocycles induced by a positive definite function on a locally compact abelian group  [ Sur les 1-cocycles induits par une fonction de type positif sur un groupe abélien localement compact ]
Annales Mathématiques Blaise Pascal, Tome 21 (2014) no. 1, pp. 61-69.

Soit ϕ une fonction de type positif normalisée sur un groupe localement compact abélien G, et π ϕ la représentation unitaire de G obtenue par construction GNS. Nous donnons des conditions nécessaires et suffisantes pour l’annulation de la 1-cohomologie H 1 (G,π ϕ ) et de la 1-cohomologie réduite H ¯ 1 (G,π ϕ ). Par exemple, H ¯ 1 (G,π ϕ )=0 si et seulement si ou bien Hom(G,)=0 ou bien μ ϕ (1 G )=0, où 1 G est le caractère trivial de G et μ ϕ est la mesure de probabilité sur le dual de Pontryagin G ^ associée à ϕ par le théorème de Bochner. Cela simplifie un argument de Guichardet (Théorème 4 de [7]).

For ϕ a normalized positive definite function on a locally compact abelian group G, let π ϕ be the unitary representation associated to ϕ by the GNS construction. We give necessary and sufficient conditions for the vanishing of 1-cohomology H 1 (G,π ϕ ) and reduced 1-cohomology H ¯ 1 (G,π ϕ ). For example, H ¯ 1 (G,π ϕ )=0 if and only if either Hom(G,)=0 or μ ϕ (1 G )=0, where 1 G is the trivial character of G and μ ϕ is the probability measure on the Pontryagin dual G ^ associated to ϕ by Bochner’s Theorem. This streamlines an argument of Guichardet (see Theorem 4 in [7]).

DOI : https://doi.org/10.5802/ambp.335
Classification : 43A35
Mots clés : continuous 1-cohomology, cyclic representation, GNS construction, locally compact abelian group, positive definite function
@article{AMBP_2014__21_1_61_0,
     author = {Franks, Jordan and Valette, Alain},
     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 = {www.numdam.org/item/AMBP_2014__21_1_61_0/}
}
Franks, Jordan; Valette, Alain. On $1$-cocycles induced by a positive definite function on a locally compact abelian group. Annales Mathématiques Blaise Pascal, Tome 21 (2014) no. 1, pp. 61-69. doi : 10.5802/ambp.335. http://www.numdam.org/item/AMBP_2014__21_1_61_0/

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