Zappa-Sz\'ep products of semigroups and their C*-algebras

TL;DR

This paper develops a unifying framework for associating C*-algebras to Zappa-Szép products of semigroups, encompassing various known classes and providing explicit algebraic presentations.

Contribution

It introduces a construction of C*-algebras for Zappa-Szép products, generalizing existing classes and explicitly describing their algebraic structure.

Findings

Unified framework for Zappa-Szép product C*-algebras

Explicit presentations of associated C*-algebras

Application to classical examples like Baumslag-Solitar groups

Abstract

Zappa-Sz\'ep products of semigroups encompass both the self-similar group actions of Nekrashevych and the quasi-lattice-ordered groups of Nica. We use Li's construction of semigroup $C^*$-algebras to associate a $C^*$-algebra to Zappa-Sz\'ep products and give an explicit presentation of the algebra. We then define a quotient $C^*$-algebra that generalises the Cuntz-Pimsner algebras for self-similar actions. We indicate how known examples, previously viewed as distinct classes, fit into our unifying framework. We specifically discuss the Baumslag-Solitar groups, the binary adding machine, the semigroup $\mathbb{N}\rtimes\mathbb{N}^\times$, and the $ax+b$-semigroup $\mathbb{Z}\rtimes\mathbb{Z}^\times$.