Simplicity of $*$-algebras of non-Hausdorff $\mathbb{Z}_2$-multispinal groupoids

TL;DR

This paper investigates the simplicity of $C^*$-algebras from self-similar $bZ_2$-multispinal groupoids, generalizing known results and establishing new sufficient conditions for simplicity based on groupoid behavior and combinatorial designs.

Contribution

It introduces new sufficient conditions for the simplicity of Steinberg algebras from self-similar $bZ_2$-multispinal groupoids, extending previous work and identifying novel combinatorial structures.

Findings

Established a sufficient condition for algebraic simplicity based on groupoid properties.

Identified an infinite family of combinatorial designs relevant to the algebraic structure.

Proved the associated $C^*$-algebras are simple under these conditions.

Abstract

We study simplicity of $C^*$-algebras arising from self-similar groups of $\mathbb{Z}_2$-multispinal type, a generalization of the Grigorchuk case whose simplicity was first proved by L. Clark, R. Exel, E. Pardo, C. Starling, and A. Sims in 2019, and we prove results generalizing theirs. Our first main result is a sufficient condition for simplicity of the Steinberg algebra satisfying conditions modeled on the behavior of the groupoid associated to the first Grigorchuk group. This closely resembles conditions found by B. Steinberg and N. Szak\'acs. As a key ingredient we identify an infinite family of $2-(2q-1,q-1,q/2-1)$-designs, where $q$ is a positive even integer. We then deduce the simplicity of the associated $C^*$-algebra, which is our second main result. Results of similar type were considered by B. Steinberg and N. Szak\'acs in 2021, and later by K. Yoshida, but their methods did not follow the original methods of the five authors.