An algebraic analogue of Exel-Pardo C*-algebras

TL;DR

This paper develops algebraic analogues of certain $C^*$-algebras associated with matrices and self-similar actions, establishing their structure and isomorphisms with Steinberg algebras.

Contribution

It introduces algebraic versions of Katsura and Exel-Pardo $C^*$-algebras, proving key theorems and connecting them to Steinberg algebras over non-Hausdorff groupoids.

Findings

Proved a Graded Uniqueness Theorem for the algebraic $C^*$-analogues.

Constructed an isomorphism between the algebraic Exel-Pardo algebra and Steinberg algebras.

Showed that unital algebraic Katsura $C^*$-analogues are isomorphic to Steinberg algebras.

Abstract

We introduce an algebraic version of the Katsura $C^*$-algebra of a pair $A,B$ of integer matrices and an algebraic version of the Exel-Pardo $C^*$-algebra of a self-similar action on a graph. We prove a Graded Uniqueness Theorem for such algebras and construct a homomorphism of the latter into a Steinberg algebra that, under mild conditions, is an isomorphism. Working with Steinberg algebras over non-Hausdorff groupoids we prove that in the unital case, our algebraic version of Katsura $C^*$-algebras are all isomorphic to Steinberg algebras.