Josiah Aakre | Searcharxiv

arXiv2026

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.

math.RA↗

arXiv2025

Simplicity of algebras and $C^*$-algebras of self-similar groupoids

Many previously studied path algebras or self-similar group algebras may be viewed as Steinberg algebras of self-similar groupoids. By way of inverse semigroup algebras, we characterize when the Steinberg algebra of a self-similar groupoid is simple. We show that the simplicity of the reduced $C^*$-algebra of a contracting self-similar groupoid coincides with the simplicity of the Steinberg algebra. As an aside, we show that simplicity of the two algebras sometimes depends only on the skeleton of the self-similar groupoid acting on a strongly connected graph. Finally, we apply our methods to examples including a self-similar groupoid akin to multispinal self-similar groups and a self-similar groupoid built from the well-known Basilica group.

math.RA↗