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

2007/02/23 by Habib Amiri, Amiri, H.
Mathematics · #22A22 #22A25 #43A35 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.math/0702709

openalex publication_date 2007/02/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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=(G⁰,\H_u\,μ), μ is a quasi invariant measure on G⁰ and ξ∈ L^∞(G⁰,\h). [10]. In this paper firt we prove that if μ is a quasi invariant ergodic measure on G⁰, 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 μ is an ergodic quasi invariant measure on G⁰, for a positive definite function p which is an extreme point of \mP^μ₁(G) the corresponding representation L is irreducible and conversely, every irreducible representation L of G on a Hilbert bundle \h=(G⁰,\H_u\,μ) and every section ξ∈ \h(μ) with norm one, define an extreme point of \mP^μ₁(G).

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