Exel-Pardo algebras of self-similar $k$-graphs

TL;DR

This paper introduces Exel-Pardo algebras for self-similar k-graphs, establishing their key properties, uniqueness theorems, and ideal structure, with applications to Steinberg algebras.

Contribution

It defines Exel-Pardo algebras for self-similar k-graphs and proves their graded and Cuntz-Krieger uniqueness theorems, extending previous results to this new setting.

Findings

Proved graded and Cuntz-Krieger uniqueness theorems for Exel-Pardo algebras.

Presented these algebras as Steinberg algebras.

Analyzed the ideal structure of the algebras.

Abstract

We introduce the Exel-Pardo $*$-algebra $\mathrm{EP}_R(G,\Lambda)$ associated to a self-similar $k$-graph $(G,\Lambda,\varphi)$. We prove the $\mathbb{Z}^k$-graded and Cuntz-Krieger uniqueness theorems for such algebras and investigate their ideal structure. In particular, we modify the graded uniqueness theorem for self-similar 1-graphs, and then apply it to present $\mathrm{EP}_R(G,\Lambda)$ as a Steinberg algebra and to study the ideal structure.