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 "$A\subseteq B$" 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.

New results, including the addition of the converse in the first main result (now numbered 3.2)

References in corpus (1)

• Normalizers and Approximate Units for Inclusions of C*-Algebras