The Zappa-Szép product of twisted groupoids

2024/06/01 by Duwenig, Anna, Li, Boyu · 1 citation
#22A22 #46L05 #46L55 #FOS: Mathematics #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.2406.00466

Abstract

We define and study the external and the internal Zappa-Szép product of twists over groupoids. We determine when a pair (Σ₁,Σ₂) of twists over a matched pair (G₁,G₂) of groupoids gives rise to a Zappa-Szép twist Σ over the Zappa-Szép product G₁\bowtieG₂. We prove that the resulting (reduced and full) twisted groupoid C*-algebra of the Zappa-Szép twist Σ→ G₁\bowtieG₂ is a C*-blend of its subalgebras corresponding to the subtwists Σ_i→ G_i. Using Kumjian-Renault theory, we then prove a converse: Any C*-blend in which the intersection of the three algebras is a Cartan subalgebra in all of them, arises as the reduced twisted groupoid C*-algebras from such a Zappa-Szép twist Σ→ G₁\bowtieG₂ of two twists Σ₁→ G₁ and Σ₂→ G₂.

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• Split Lemma and First Isomorphism Theorem for groupoids

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