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Rigidity of Generalized Furstenberg Boundaries and Applications to Intermediate Crossed Products

Tattwamasi Amrutam, Chunlin Liu

TL;DR

The paper develops a relative boundary theory for actions of a discrete group $\Gamma$ on a compact space $X$, introducing the universal $H$-boundary $RB_X(\Gamma, H)$ and showing it is $\Gamma$-minimal and $H$-strongly proximal over $X$. When $H\leq_c \Gamma$ is commensurated and $X$ is $H$-minimal, the authors canonically identify $RB_X(\Gamma,H)$ with the generalized Furstenberg boundary $\partial_F(H,X)$ under a $\Gamma$-extension, unifying previous relative-boundary constructions. They then define $X$-plump subgroups to connect dynamics with operator-algebraic rigidity, proving that certain intermediate $C^*$-algebras between crossed products are simple and, under the approximation property, must themselves be crossed products. Overall, the work provides a robust framework linking relative boundary theory to rigidity phenomena in reduced crossed products and expands the catalog of examples beyond $C^*$-simple groups.

Abstract

We develop a relative boundary theory for actions of discrete groups on compact spaces and use it to derive rigidity results for reduced crossed products. For a discrete group $Γ$ acting on a compact space $X$ and a subgroup $H$, we construct a universal boundary over $X$ which is minimal as a $Γ$-system and strongly proximal with respect to $H$. When $H\le_cΓ$ is commensurated and the $H$-action on $X$ is minimal, we show that this universal boundary agrees, in a canonical $Γ$-equivariant way, with the generalized Furstenberg boundary of $(H,X)$, thereby unifying and extending earlier results on relative boundaries. As an application, we introduce the notion of an $X$-plump subgroup given a $Γ$-space $X$, a generalized version of plumpness tailored to crossed products. Under natural dynamical hypotheses, this leads to new examples of irreducible $C^*$-inclusions. Under additional assumptions, we also show that every intermediate $C^*$-algebra is a crossed product.

Rigidity of Generalized Furstenberg Boundaries and Applications to Intermediate Crossed Products

TL;DR

The paper develops a relative boundary theory for actions of a discrete group on a compact space , introducing the universal -boundary and showing it is -minimal and -strongly proximal over . When is commensurated and is -minimal, the authors canonically identify with the generalized Furstenberg boundary under a -extension, unifying previous relative-boundary constructions. They then define -plump subgroups to connect dynamics with operator-algebraic rigidity, proving that certain intermediate -algebras between crossed products are simple and, under the approximation property, must themselves be crossed products. Overall, the work provides a robust framework linking relative boundary theory to rigidity phenomena in reduced crossed products and expands the catalog of examples beyond -simple groups.

Abstract

We develop a relative boundary theory for actions of discrete groups on compact spaces and use it to derive rigidity results for reduced crossed products. For a discrete group acting on a compact space and a subgroup , we construct a universal boundary over which is minimal as a -system and strongly proximal with respect to . When is commensurated and the -action on is minimal, we show that this universal boundary agrees, in a canonical -equivariant way, with the generalized Furstenberg boundary of , thereby unifying and extending earlier results on relative boundaries. As an application, we introduce the notion of an -plump subgroup given a -space , a generalized version of plumpness tailored to crossed products. Under natural dynamical hypotheses, this leads to new examples of irreducible -inclusions. Under additional assumptions, we also show that every intermediate -algebra is a crossed product.
Paper Structure (7 sections, 32 theorems, 224 equations)

This paper contains 7 sections, 32 theorems, 224 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dynamics
  4. Crossed product $C^*$-algebra
  5. Universal Strongly Proximal Extension for Subgroups
  6. Relative Boundary Extension Theorem
  7. -plump

Key Result

Theorem A

Let $H\le \Gamma$. Let $X$ be a $\Gamma$-space which is $H$-minimal. Consider the class $\mathcal{C}$ of all $\Gamma$-extensions $\pi:Y\to X$ such that Morphisms in $\mathcal{C}$ are $\Gamma$-equivariant continuous maps $f:Y\to Y'$ satisfying $\pi' \circ f=\pi$. Then there exists a universal object $\pi_{RB}: RB_X(\Gamma,H)\to X$ in $\mathcal{C}$: for every $\pi:Y\to X$ in $\mathcal{C}$, there ex

Theorems & Definitions (86)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 76 more