On $1$-cocycles induced by a positive definite function on a locally compact abelian group

Franks, Jordan, and Valette, Alain. "On $1$-cocycles induced by a positive definite function on a locally compact abelian group." Annales mathématiques Blaise Pascal 21.1 (2014): 61-69. <http://eudml.org/doc/275522>.

@article{Franks2014,
abstract = {For $\varphi $ a normalized positive definite function on a locally compact abelian group $G$, let $\pi _\varphi $ be the unitary representation associated to $\varphi $ by the GNS construction. We give necessary and sufficient conditions for the vanishing of 1-cohomology $H^1(G,\pi _\varphi )$ and reduced 1-cohomology $\overline\{H\}^1(G,\pi _\varphi )$. For example, $\overline\{H\}^1(G,\pi _\varphi )=0$ if and only if either $\text\{Hom\}(G,\mathbb\{C\})=0$ or $\mu _\varphi (1_G)=0$, where $1_G$ is the trivial character of $G$ and $\mu _\varphi $ is the probability measure on the Pontryagin dual $\hat\{G\}$ associated to $\varphi $ by Bochner’s Theorem. This streamlines an argument of Guichardet (see Theorem 4 in [7]).},
affiliation = {Mathematisches Institut Universität Bonn Endenicher Allee 60 53115 Bonn Germany; Institut de Mathématiques Université de Neuchâtel Unimail, 11 Rue Emile Argand CH-2000 Neuchâtel Switzerland},
author = {Franks, Jordan, Valette, Alain},
journal = {Annales mathématiques Blaise Pascal},
keywords = {continuous 1-cohomology; cyclic representation; GNS construction; locally compact abelian group; positive definite function; GNS-contruction},
language = {eng},
month = {1},
number = {1},
pages = {61-69},
publisher = {Annales mathématiques Blaise Pascal},
title = {On $1$-cocycles induced by a positive definite function on a locally compact abelian group},
url = {http://eudml.org/doc/275522},
volume = {21},
year = {2014},
}

TY - JOUR
AU - Franks, Jordan
AU - Valette, Alain
TI - On $1$-cocycles induced by a positive definite function on a locally compact abelian group
JO - Annales mathématiques Blaise Pascal
DA - 2014/1//
PB - Annales mathématiques Blaise Pascal
VL - 21
IS - 1
SP - 61
EP - 69
AB - For $\varphi $ a normalized positive definite function on a locally compact abelian group $G$, let $\pi _\varphi $ be the unitary representation associated to $\varphi $ by the GNS construction. We give necessary and sufficient conditions for the vanishing of 1-cohomology $H^1(G,\pi _\varphi )$ and reduced 1-cohomology $\overline{H}^1(G,\pi _\varphi )$. For example, $\overline{H}^1(G,\pi _\varphi )=0$ if and only if either $\text{Hom}(G,\mathbb{C})=0$ or $\mu _\varphi (1_G)=0$, where $1_G$ is the trivial character of $G$ and $\mu _\varphi $ is the probability measure on the Pontryagin dual $\hat{G}$ associated to $\varphi $ by Bochner’s Theorem. This streamlines an argument of Guichardet (see Theorem 4 in [7]).
LA - eng
KW - continuous 1-cohomology; cyclic representation; GNS construction; locally compact abelian group; positive definite function; GNS-contruction
UR - http://eudml.org/doc/275522
ER -