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

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 $C^{\ast }$-algebras to associate a $C^{\ast }$-algebra to Zappa-Sz'ep products and give an explicit presentation of the algebra. We then define a quotient $C^{\ast }$-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 $\mathbb{N}⋊{\mathbb{N}}^{\times }$, and the $ax+b$-semigroup $\mathbb{Z}⋊{\mathbb{Z}}^{\times }$.