$C^*$-simplicity and Ozawa conjecture for groupoid $C^*$-algebras, part I: injective envelopes

TL;DR

This paper explores the boundaries of groupoid dynamical systems, establishing the equivalence of Hamana and Furstenberg boundaries for discrete groupoids, and relating injective envelopes to crossed products.

Contribution

It introduces the concept of the Hamana boundary for groupoids, proves its equivalence to the Furstenberg boundary, and connects injective envelopes with crossed product structures.

Findings

Hamana and Furstenberg boundaries coincide for discrete groupoids

Established the existence of a minimal boundary for groupoids

Linked injective envelopes with reduced crossed products

Abstract

This paper studies injective envelopes of groupoid dynamical systems and the corresponding boundaries. Analogue to the group case, we associate a bundle of compact Hausdorff spaces to any (discrete) groupoid (the Hamana boundary of the groupoid). We study bundles of compact topological spaces equipped with an action of a groupoid. We show that any groupoid has a minimal boundary (the Furstenberg boundary of the groupoid). We prove that the Hamana and Furstenberg boundaries are the same, for (discrete) groupoids. We find the relation between the reduced crossed product of the $\G$-injective envelop of a groupoid dynamical system and the injective envelope of the reduced crossed product of the original system.