On finitely aligned left cancellative small categories, Zappa-Sz\'ep products and Exel-Pardo algebras

TL;DR

This paper explores the construction and analysis of Toeplitz and Cuntz-Krieger $C^*$-algebras from finitely aligned left cancellative small categories, focusing on Zappa-Szép products and their application to Exel-Pardo algebras in graph theory.

Contribution

It introduces a new approach to Exel-Pardo algebras via Zappa-Szép products of categories and groups, expanding methods for constructing $C^*$-algebras from small categories.

Findings

Develops a framework for Toeplitz and Cuntz-Krieger algebras from small categories.

Provides a novel approach to Exel-Pardo algebras for row-finite graphs.

Discusses relationships between different $C^*$-algebra constructions.

Abstract

We consider Toeplitz and Cuntz-Krieger $C^*$-algebras associated with finitely aligned left cancellative small categories. We pay special attention to the case where such a category arises as the Zappa-Sz\'ep product of a category and a group linked by a one-cocycle. As our main application, we obtain a new approach to Exel-Pardo algebras in the case of row-finite graphs. We also present some other ways of constructing $C^*$-algebras from left cancellative small categories and discuss their relationship.