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

If G is a locally compact groupoid with a Haar system λ, then a positive definite function p on G has a form p(x) = 〈L(x)ξ(d(x)), ξ(r(x))〉, where L is a representation of G on a Hilbert bundle H = (G, {Hu}, μ), μ is a quasi invariant measure on G 0 and ξ ∈ L(G,H). [10]. In this paper firt we prove that if μ is a quasi invariant ergodic measure on G, then two corresponding representations of G and Cc(G) are irreducible in the same time. Then by using the theory of positive linear functionals on C∗(G) we show that when μ is an ergodic quasi invariant measure on G, for a positive definite function p which is an extreme point of Pμ 1 (G) the corresponding representation L is irreducible and conversely, every irreducible representation L of G on a Hilbert bundle H = (G, {Hu}, μ) and every section ξ ∈ H(μ) with norm one, define an extreme point of P μ 1 (G).