[Paper Review] Zappa-Szep products of semigroups and their C*-algebras

Nathan Brownlowe, Jacqui Ramagge+2 more|arXiv (Cornell University)|Sep 2, 2013

semigroups and automata theoryComputer Science23 references6 citations

TL;DR

This paper introduces a unified framework for constructing C*-algebras from Zappa-Szép products of semigroups, using Li's semigroup C*-algebra construction. It presents explicit generators-and-relations presentations for both the full C*-algebra and a boundary quotient, showing that known classes—including quasi-lattice ordered groups, self-similar group actions, and Baumslag-Solitar groups—arise as special cases. The key contribution is a generalization that unifies Cuntz-Pimsner algebras and Nica's C*-algebras under a single algebraic structure.

ABSTRACT

Zappa-Szép 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-Szép 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 $\mathbb{N} times\mathbb{N}^ imes$, and the $ax+b$-semigroup $\mathbb{Z} times\mathbb{Z}^ imes$.

Motivation & Objective

• To unify disparate classes of C*-algebras—specifically those from quasi-lattice ordered groups and self-similar group actions—under a single framework.
• To extend Li's semigroup C*-algebra construction to Zappa-Szép products of left-cancellative semigroups with identities.
• To define and analyze a boundary quotient C*-algebra that generalizes Cuntz-Pimsner algebras for self-similar actions.
• To provide explicit generators-and-relations presentations for both the full C*-algebra and its boundary quotient, enhancing tractability.

Proposed method

• Construct the Zappa-Szép product of two left-cancellative semigroups with identities, ensuring the resulting semigroup is also left-cancellative and right LCM.
• Apply Li's construction to associate a full C*-algebra to the Zappa-Szép product, using isometric representations and projections indexed by right ideals.
• Introduce a boundary quotient C*-algebra via a quotient by an ideal generated by projections associated with foundation sets in the semigroup.
• Establish isomorphisms between the resulting C*-algebras and known algebras (e.g., Cuntz-Pimsner algebras, Nica's C*(G,P)) by verifying universal relations.
• Use induction and properties of foundation sets to prove that certain products of projections in the quotient algebra vanish, enabling the identification of the boundary quotient.
• Verify that the universal relations of known algebras (e.g., Q2, O(G,X)) match those of the constructed C*-algebras via explicit homomorphisms and surjectivity arguments.

Experimental results

Research questions

• RQ1 Can Zappa-Szép products of semigroups serve as a unifying framework for diverse C*-algebra constructions, including those from quasi-lattice ordered groups and self-similar group actions?
• RQ2 How can Li's semigroup C*-algebra construction be adapted to Zappa-Szép products to yield a full C*-algebra with an explicit presentation?
• RQ3 What is the structure of the boundary quotient C*-algebra in this context, and how does it relate to Cuntz-Pimsner algebras?
• RQ4 Do known examples such as the 2-adic ring C*-algebra Q2 and the Cuntz-Pimsner algebra O(G,X) arise naturally as quotients of the full C*-algebra of a Zappa-Szép product?
• RQ5 Can the framework be extended to products of self-similar actions, and what properties (e.g., simplicity) can be deduced?

Key findings

• The full C*-algebra of a Zappa-Szép product of left-cancellative semigroups admits an explicit presentation via generators and relations, generalizing Li's construction.
• The boundary quotient C*-algebra is isomorphic to the Cuntz-Pimsner algebra O(G,X) for any self-similar group action (G,X), unifying this class of algebras.
• The 2-adic ring C*-algebra Q2 is isomorphic to the boundary quotient of the Zappa-Szép product X* ✨ N, where X={0,1} and N is the adding machine.
• For the semigroup N ✨ N*, the full C*-algebra is isomorphic to Nica's C*(BS(1,2), BS(1,2)+), and the boundary quotient is isomorphic to the Cuntz-Pimsner algebra O(Z,X).
• The framework generalizes to products of self-similar actions: when the semigroup F+_θ is right LCM, the C*-algebra C*(F+_θ ✨ G) is universal for a Toeplitz-Cuntz-Krieger family and unitary representations satisfying the self-similarity relation.
• The proof of the isomorphism between the boundary quotient and O(G,X) relies on showing that the ideal generated by 1 - ∑x∈X vtxv*x equals the ideal generated by products of projections (1 - t_w t*_w) over foundation sets, using induction on word length.