On the irreducibility of induced representations of some groupoid $$C^*$$-algebras

Abstract

Let $${\mathcal {G}}$$ be a topological locally compact Hausdorff and second countable groupoid with a Haar system and $${\mathcal {K}}$$ a proper subgroupoid of $${\mathcal {G}}$$ with a Haar system too. $$({\mathcal {G}},{\mathcal {K}})$$ is an internally Gelfand pair if for any u in the unit space, the pair of isotropy groups $$({\mathcal {G}}(u), {\mathcal {K}}(u))$$ is a Gelfand pair. In this work, we prove, when $$({\mathcal {G}},{\mathcal {K}})$$ is an internally Gelfand pair that any representation of $${\mathcal {C}}^{*} ({\mathcal {G}}, {\mathcal {K}},\lambda , \alpha )$$ induced from a positive definite spherical function on an isotropy group is irreducible. We introduce also the notion of spherical bundle and after identifying it with the spectrum of some $$C^*$$ -algebra, we obtain some properties of its topology.