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

TL;DR

This paper characterizes exotic C*-algebras arising from twisted, principal étale groupoids and their abelian subalgebras, providing a streamlined description of Kumjian's C*-diagonals and conditions for trivial opaque ideals.

Contribution

It offers a precise characterization of exotic C*-algebras with abelian subalgebras and clarifies the structure of Kumjian's C*-diagonals using the extension property and opaque ideals.

Findings

Exotic C*-algebras are characterized by abelian, regular subalgebras with the extension property.

When the larger algebra is nuclear, the opaque ideal is trivial.

Kumjian's C*-diagonals are identified as regular abelian subalgebras with the extension property and vanishing opaque ideal.

Abstract

We characterize exotic C*-algebras of twisted, principal \'etale 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.