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Stable rank one in nonnuclear crossed products

Jamie Bell, Shirly Geffen, David Kerr

TL;DR

The paper studies stable rank one for simple nonnuclear crossed products arising from minimal actions of free groups on the Cantor set. It introduces square divisibility (and a Cantor-set variant) as a dynamical mechanism that enables unitary rotations, perturbations, and boundary-control arguments to realize nilpotent perturbations, leading to stable rank one via Rørdam’s strategy. By exploiting URP and dynamical comparison in amenable settings, and developing a diagonal-action machine for free products, the authors prove generic stable rank one for reduced crossed products associated to generic actions in WA(G*H,X) and A*(F_d,X), and show that certain conjugacy classes are meagre. They further demonstrate permanence results for product actions and provide explicit squarely divisible action examples, illustrating the reach of the approach beyond the reduced group C*-algebras. Overall, the work ties amenability-based dynamical techniques to structural regularity properties in nonnuclear crossed products, with broad implications for classification-type regularity and dynamical-C*-algebra interactions.

Abstract

We initiate an investigation into the local structure of simple nonnuclear C$^*$-crossed products by showing that stable rank one is generic within two natural classes of minimal actions of free groups on the Cantor set. The arguments also apply to some other free product groups. Our approach is inspired by Li and Niu's stable rank one theorem in the amenable setting and also yields a streamlined argument in that case, along with a generalization to product actions.

Stable rank one in nonnuclear crossed products

TL;DR

The paper studies stable rank one for simple nonnuclear crossed products arising from minimal actions of free groups on the Cantor set. It introduces square divisibility (and a Cantor-set variant) as a dynamical mechanism that enables unitary rotations, perturbations, and boundary-control arguments to realize nilpotent perturbations, leading to stable rank one via Rørdam’s strategy. By exploiting URP and dynamical comparison in amenable settings, and developing a diagonal-action machine for free products, the authors prove generic stable rank one for reduced crossed products associated to generic actions in WA(G*H,X) and A*(F_d,X), and show that certain conjugacy classes are meagre. They further demonstrate permanence results for product actions and provide explicit squarely divisible action examples, illustrating the reach of the approach beyond the reduced group C*-algebras. Overall, the work ties amenability-based dynamical techniques to structural regularity properties in nonnuclear crossed products, with broad implications for classification-type regularity and dynamical-C*-algebra interactions.

Abstract

We initiate an investigation into the local structure of simple nonnuclear C-crossed products by showing that stable rank one is generic within two natural classes of minimal actions of free groups on the Cantor set. The arguments also apply to some other free product groups. Our approach is inspired by Li and Niu's stable rank one theorem in the amenable setting and also yields a streamlined argument in that case, along with a generalization to product actions.

Paper Structure

This paper contains 12 sections, 62 theorems, 191 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notational conventions and preliminaries
  3. Square divisibility
  4. Stable rank one
  5. Square divisibility and amenable groups
  6. Square divisibility and product actions
  7. A diagonal action machine
  8. Spaces of actions
  9. Generic square divisibility I
  10. Generic square divisibility II
  11. Meagreness of orbits: proof of Theorem \ref{['T-meagre']}
  12. Examples of squarely divisible actions of free groups

Key Result

Theorem A

Let $G$ be a residually finite countable discrete group and $H$ an amenable countable discrete group containing a normal infinite cyclic subgroup. Let $X$ be the Cantor set. For a generic action in $\mathop{\mathrm{WA}}\nolimits (G*H,X)$ the reduced crossed product $C(X)\rtimes_\lambda (G * H)$ has

Figures (3)

  • Figure 1: $(O_1,O_2,E)$-square divisibility ($n=3$) in the Cantor set setting with $U = \emptyset$ (as in Proposition \ref{['P-SD0Dim']}) and arrows indicating subequivalences.
  • Figure 2: Castles ${\mathscr C}_1 \prec {\mathscr C}_2$ for $T$ with $\mathop{\mathrm{base}}\nolimits ({\mathscr C}_1 ) = A_1 \sqcup A_2 \sqcup A_3$ and $\mathop{\mathrm{base}}\nolimits ({\mathscr C}_2 ) = B_1 \sqcup B_2 \sqcup B_3$.
  • Figure 3: The initial paths of length two in the ordered Bratteli diagram associated to Figure \ref{['F-partition']}.

Theorems & Definitions (136)

  • Theorem A
  • Corollary B
  • Theorem C
  • Theorem D
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Definition 3.1
  • Definition 3.2
  • ...and 126 more