Characterize UCT Kirchberg algebras admitting C*-diagonals

Determine precisely which Kirchberg algebras that satisfy the universal coefficient theorem admit a C*-diagonal, that is, a Cartan subalgebra whose associated Renault groupoid is principal. Provide a complete characterization of the class of UCT Kirchberg algebras containing such C*-diagonals.

Background

Cartan subalgebras of C*-algebras correspond to twisted étale groupoid models, and C*-diagonals are the special case corresponding to principal groupoids. Many known constructions yield topologically principal (but not principal) groupoids, producing Cartan subalgebras that are not C*-diagonals.

While it is known that every UCT Kirchberg algebra has a Cartan subalgebra via topologically principal groupoid models, the existence of C*-diagonals is subtler. There are positive results in specific cases (e.g., Hjelmborg for O2, constructions via amenable actions yielding further examples, and K-theoretic criteria by Brown–Clark–Sierakowski–Sims), but no general characterization is known.

This paper advances the landscape by proving that Ok contains a C*-diagonal with Cantor spectrum for all finite k≥2 and by constructing uncountably many UCT Kirchberg algebras with C*-diagonals. Nonetheless, the full classification of which UCT Kirchberg algebras admit C*-diagonals remains unresolved.