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 $G$.
Highlights
The Gel’fand-Naimark-Segal construction provides a correspondence between positive definite functions φ on a locally compactKeywords: continuous 1-cohomology, cyclic representation, GNS construction, locally compact abelian group, positive definite function.Math. classification: 43A35.J
Φ is an extreme point in the cone P(G) of positive definite functions on G if and only if πφ is an irreducible representation; or, there exists a constant a > 0 such that φ−a is again positive definite if and only if πφ has nonzero fixed vectors
Without relying on the cohomological machinery available in the literature, we achieve by completely elementary means the results of this paper, namely, we show that the existence of nontrivial 1-cohomology is determined by two factors: the existence of non-trivial homomorphisms from G to C and the behavior of μφ near the trivial character 1G
Summary
The Gel’fand-Naimark-Segal construction (see [1]) provides a correspondence between positive definite functions φ on a locally compact. Φ is an extreme point in the cone P(G) of positive definite functions on G if and only if πφ is an irreducible representation; or, there exists a constant a > 0 such that φ−a is again positive definite if and only if πφ has nonzero fixed vectors (see [4]) In view of their importance for rigidity questions and Kazhdan’s property (T), it seems natural to try to fit 1-cohomology and reduced 1cohomology of πφ in that dictionary. Without relying on the cohomological machinery available in the literature (see [8, 1]), we achieve by completely elementary means the results of this paper, namely, we show that the existence of nontrivial 1-cohomology is determined by two factors: the existence of non-trivial homomorphisms from G to C and the behavior of μφ near the trivial character 1G This is all delineated in a precise way in Theorem 1.
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