Relations between positive definite functions and irreducible representations on a locally compact groupoid

Abstract

If $G$ is a locally compact groupoid with a Haar system $\lambda$, then a positive definite function $p$ on $G$ has a form $p(x)=< L(x)\xi(d(x)),\xi(r(x))>$, where $L$ is a representation of $G$ on a Hilbert bundle ${\h}=(G^0,\{H_u\},\mu)$, $\mu$ is a quasi invariant measure on $G^0$ and $\xi\in L^{\infty}(G^0,{\h})$. [10]. In this paper firt we prove that if $\mu$ is a quasi invariant ergodic measure on $G^0$, then two corresponding representations of $G$ and $C_c(G)$ are irreducible in the same time. Then by using the theory of positive linear functionals on $C^*(G)$ we show that when $\mu$ is an ergodic quasi invariant measure on $G^0$, for a positive definite function $p$ which is an extreme point of ${\mP}^{\mu}_1(G)$ the corresponding representation $L$ is irreducible and conversely, every irreducible representation $L$ of $G$ on a Hilbert bundle ${\h}=(G^0,\{H_u\},\mu)$ and every section $\xi\in {\h}(\mu)$ with norm one, define an extreme point of ${\mP}^{\mu}_1(G)$.