Zappa–Szép products of semigroups and their C⁎C⁎-algebras

Zappa–Szép products of semigroups provide a rich class of examples of semigroups that include the self-similar group actions of Nekrashevych. We use Li's construction of semigroup C⁎C⁎-algebras to associate a C⁎C⁎-algebra to Zappa–Szép products and give an explicit presentation of the algebra. We then define a quotient C⁎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 N⋊N×N⋊N×, and the ax+bax+b-semigroup Z⋊Z×Z⋊Z×.