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.

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]).

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]).

DOI: 10.5802/ambp.335
Classification: 43A35
Keywords: continuous 1-cohomology, cyclic representation, GNS construction, locally compact abelian group, positive definite function
Jordan Franks 1; Alain Valette 2

1 Mathematisches Institut Universität Bonn Endenicher Allee 60 53115 Bonn Germany
2 Institut de Mathématiques Université de Neuchâtel Unimail, 11 Rue Emile Argand CH-2000 Neuchâtel Switzerland
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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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