Continuous-trace groupoid 𝐶*-algebras. III

Abstract

Suppose that G \mathcal {G} is a second countable locally compact groupoid with a Haar system and with abelian isotropy. We show that the groupoid C ∗ C^{*} -algebra C ∗ ( G , λ ) C^*(\mathcal {G},\lambda ) has continuous trace if and only if there is a Haar system for the isotropy groupoid A \mathcal {A} and the action of the quotient groupoid G / A \mathcal {G}/\mathcal {A} is proper on the unit space of G \mathcal {G} .