[Paper Review] The Furstenberg Boundary of a Groupoid

Clemens Borys|arXiv (Cornell University)|Apr 22, 2019

Advanced Operator Algebra Research19 references4 citations

TL;DR

This paper introduces the Furstenberg boundary for locally compact Hausdorff étale groupoids with compact unit space by constructing a groupoid-equivariant injective envelope, generalizing the classical Furstenberg boundary of discrete groups. It establishes that the absence of recurrent amenable subgroups in the isotropy implies the intersection property, leading to a sufficient criterion for C*-simplicity of minimal groupoids.

ABSTRACT

We define the Furstenberg boundary of a locally compact Hausdorff étale groupoid, generalising the Furstenberg boundary for discrete groups, by providing a construction of a groupoid-equivariant injective envelope. Using this injective envelope, we establish the absence of recurrent amenable subgroups in the isotropy as a sufficient criterion for the intersection property of a locally compact Hausdorff étale groupoid with compact unit space and no fixed points. This yields a criterion for C*-simplicity of minimal groupoids.

Motivation & Objective

• To generalize the Furstenberg boundary from discrete groups to locally compact Hausdorff étale groupoids with compact unit space.
• To construct a groupoid-equivariant injective envelope as a foundational tool for boundary theory in groupoid C*-algebras.
• To establish a sufficient condition for the intersection property in étale groupoids using isotropy structure.
• To derive a criterion for C*-simplicity of minimal étale groupoids based on the absence of recurrent amenable subgroups in isotropy.

Proposed method

• Introduces an induction functor to transport injective operator systems into the category of operator systems with groupoid actions.
• Constructs the boundary as the spectrum of the groupoid-equivariant injective envelope of the trivial algebra C.
• Uses the topology of the space of closed subgroups of isotropy groups to define recurrence of subgroups.
• Applies the theory of injective envelopes in dynamical categories to analyze the structure of the boundary groupoid.
• Analyzes the isotropy map Φ: ˜X → Sub(G) to prove continuity and closedness of orbits under conjugation.
• Leverages the fact that trivial isotropy in the boundary groupoid implies principality and hence the intersection property.

Experimental results

Research questions

• RQ1 How can the Furstenberg boundary be generalized from discrete groups to étale groupoids?
• RQ2 What conditions on the isotropy of a groupoid ensure the intersection property in its boundary?
• RQ3 Does the absence of recurrent amenable subgroups in isotropy imply the intersection property for the groupoid?
• RQ4 Can the intersection property be used to characterize C*-simplicity in minimal étale groupoids?
• RQ5 Is the residual intersection property equivalent to the absence of recurrent amenable subgroups in isotropy?

Key findings

• The Furstenberg boundary of a locally compact Hausdorff étale groupoid is constructed via a groupoid-equivariant injective envelope.
• The boundary groupoid ˜G is principal if and only if the original groupoid G has no recurrent amenable subgroups in its isotropy.
• The absence of recurrent amenable subgroups in isotropy implies the intersection property for G, provided all orbits have at least two points.
• For minimal étale groupoids with compact unit space, the absence of recurrent amenable subgroups in isotropy implies C*-simplicity.
• The isotropy map Φ: ˜X → Sub(G) is continuous, ensuring the orbit of isotropy subgroups is closed, which is key to proving principality of the boundary groupoid.
• The result extends Kennedy's characterization of C*-simplicity to the groupoid setting, providing a geometric criterion based on isotropy structure.