On Kumjian's C*-diagonals and the opaque ideal

Abstract

We characterize exotic C*-algebras of twisted, principal, étale groupoids, together with the abelian subalgebra associated to the unit space, as precisely being the inclusions "A ⊆ B" of C*-algebras in which A is abelian, regular, and satisfies the extension property (pure states extend uniquely to B). When B is moreover nuclear, we deduce that the corresponding opaque ideal is trivial. As an application, we give a streamlined characterization of Kumjian's C*-diagonals as the regular abelian subalgebras satisfying the extension property with vanishing opaque ideal.