Koopman representations for positive definite functions

Abstract

We show that for any locally compact second countable group $G$ and any continuous positive definite function $\phi:G\rightarrow\mathbb{C}$, there exists an ergodic measure preserving system $(X,\mathscr{B},\mu,\{T_g\}_{g \in G})$ and a function $f \in L^2(X,\mu)$ for which $\phi(g) = \langle T_gf,f\rangle$. We also show that if $G$ is a countably infinite abelian group, then there exists a (not necessarily ergodic) measure preserving system $(X,\mathscr{B},\mu,\{T_g\}_{g \in G})$ and a function $f \in L^2(X,\mu)$ with $|f| = \phi(0)$ and $\phi(g) = \langle T_gf,f\rangle$.