Zappa-Szép products of semigroups and their \(C^\ast\)-algebras. (English) Zbl 1286.22002
Summary: 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 \(C^\ast\)-algebras to associate a \(C^\ast\)-algebra to Zappa-Szép 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 \(\mathbb N \rtimes \mathbb N^{\times}\), and the \(ax+b\)-semigroup \(\mathbb Z \rtimes \mathbb Z^{\times}\).
MSC:
| 22A30 | Other topological algebraic systems and their representations |
| 22D25 | \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations |
| 43A07 | Means on groups, semigroups, etc.; amenable groups |
| 46L05 | General theory of \(C^*\)-algebras |
Keywords:
\(C^\ast\)-algebra; semigroup; Zappa-Szép product; self-similar group; Baumslag-Solitar group; quasi-lattice ordered groupReferences:
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