Koopman representations for positive definite functions

Abstract

We show that for any locally compact second countable group $G$ and any continuous positive definite function $\ddot{I}•:G\to \mathbb{C}$, there exists an ergodic measure preserving system $(X,\mathcal{B},\hat{I}¼,\{T_g{\}}_{g\in G})$ and a function $f\in L^2(X,\hat{I}¼)$ for which $\ddot{I}•(g)=⟨T_{g}f,f⟩$. We also show that if $G$ is a countably infinite abelian group, then there exists a (not necessarily ergodic) measure preserving system $(X,\mathcal{B},\hat{I}¼,\{T_g{\}}_{g\in G})$ and a function $f\in L^2(X,\hat{I}¼)$ with $|f|=\ddot{I}•(0)$ and $\ddot{I}•(g)=⟨T_{g}f,f⟩$.

This article is now superseded by arXiv:2409.00806