On the simplicity of Katsura algebras

We give a complete characterization of the (purely infinite) simplicity of Katsura algebras and the associated Steinberg algebras. This is achieved by characterizing when the singular ideals vanish via the self-similar groupoid model derived from Exel and Pardo. Analogous results are given for the algebras arising from the faithful quotient of the self-similar action. Finally, we describe polynomial-space algorithms to determine if each singular ideal vanishes and provide the first non-Hausdorff examples of non-contracting self-similar groupoids for which simplicity is algorithmically decidable.