Continuous-trace groupoid $C^{\displaystyle *}$-algebras. III

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