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

TL;DR

This paper investigates the conditions under which 1-cohomology and reduced 1-cohomology vanish for unitary representations derived from positive definite functions on locally compact abelian groups, linking cohomological properties to measure-theoretic conditions.

Contribution

It provides necessary and sufficient conditions for the vanishing of 1-cohomology and reduced 1-cohomology in this setting, connecting representation theory, harmonic analysis, and cohomology.

Findings

$ar{H}^1(G, ho)=0$ iff $Hom(G,C)=0$ or $ ext{measure}( ext{trivial character})=0$

Characterization of cohomology vanishing via measure and homomorphism conditions

Linking cohomological properties to measure-theoretic and algebraic structures

Abstract

For $\varphi$ a normalized positive definite function on a locally compact abelian group $G$, we consider on the one hand the unitary representation $\pi_\varphi$ associated to $\varphi$ by the GNS construction, on the other hand the probability measure $\mu_\varphi$ on the Pontryagin dual $\hat{G}$ provided by Bochner's theorem. We give necessary and sufficient conditions for the vanishing of 1-cohomology $H^1(G,\pi_\varphi)$ and reduced 1-cohomology $\bar{H}^1(G,\pi_\varphi)$. For example, $\bar{H}^1(G,\pi_\varphi)=0$ if and only if either $Hom(G,\mathbb{C})=0$ or $\mu_\varphi(1_G)=0$, where $1_G$ is the trivial character of $G$.