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Nuclear dimension of groupoid C*-algebras with large abelian isotropy, with applications to C*-algebras of directed graphs and twists

Astrid an Huef, Dana P. Williams

TL;DR

This work develops a unified framework to bound the nuclear dimension of groupoid C*-algebras with large abelian isotropy, by linking subhomogeneity, the quasi-orbit map, and dynamic asymptotic dimension. For étale groupoids with continuously varying, subhomogeneous isotropy, the authors derive a bound on $\dim_{ ext{nuc}}^{+1}(C^{*}(G))$ that factors the unit-space dimension, the dynamic asymptotic dimension of the isotropy-quotient, and the maximal dimension of isotropy representations. They apply these results to directed graphs and twists, showing that all stably finite graph C*-algebras have nuclear dimension at most 1 and giving analogous bounds for twisted groupoid C*-algebras. The paper thus advances the classification program by providing finite nuclear-dimension guarantees in natural groupoid settings and offers new tools for analyzing graph algebras via quotient constructions. Overall, it yields significant progress in understanding how isotropy and dynamic-asymptotic-dimension data control the regularity properties of groupoid C*-algebras.

Abstract

We characterise when the C*-algebra $C^*(G)$ of a locally compact and Hausdorff groupoid $G$ is subhomogeneous, that is, when its irreducible representations have bounded finite dimension; if so we establish a bound for its nuclear dimension in terms of the topological dimensions of the unit space of the groupoid and the spectra of the primitive ideal spaces of the isotropy subgroups. For an étale groupoid $G$, we also establish a bound on the nuclear dimension of its C*-algebra provided the quotient of $G$ by its isotropy subgroupid has finite dynamic asymptotic dimension in the sense of Guentner, Willet and Yu. Our results generalise those of C.~Böncicke and K.~Li to groupoids with large isotropy, including graph groupoids of directed graphs. We find that all graph C*-algebras that are stably finite have nuclear dimension at most $1$. We also show that the nuclear dimension of the C*-algebra of a twist over $G$ has the same bound on the nuclear dimension as for $C^*(G)$ and the twisted groupoid C*-algebra.

Nuclear dimension of groupoid C*-algebras with large abelian isotropy, with applications to C*-algebras of directed graphs and twists

TL;DR

This work develops a unified framework to bound the nuclear dimension of groupoid C*-algebras with large abelian isotropy, by linking subhomogeneity, the quasi-orbit map, and dynamic asymptotic dimension. For étale groupoids with continuously varying, subhomogeneous isotropy, the authors derive a bound on that factors the unit-space dimension, the dynamic asymptotic dimension of the isotropy-quotient, and the maximal dimension of isotropy representations. They apply these results to directed graphs and twists, showing that all stably finite graph C*-algebras have nuclear dimension at most 1 and giving analogous bounds for twisted groupoid C*-algebras. The paper thus advances the classification program by providing finite nuclear-dimension guarantees in natural groupoid settings and offers new tools for analyzing graph algebras via quotient constructions. Overall, it yields significant progress in understanding how isotropy and dynamic-asymptotic-dimension data control the regularity properties of groupoid C*-algebras.

Abstract

We characterise when the C*-algebra of a locally compact and Hausdorff groupoid is subhomogeneous, that is, when its irreducible representations have bounded finite dimension; if so we establish a bound for its nuclear dimension in terms of the topological dimensions of the unit space of the groupoid and the spectra of the primitive ideal spaces of the isotropy subgroups. For an étale groupoid , we also establish a bound on the nuclear dimension of its C*-algebra provided the quotient of by its isotropy subgroupid has finite dynamic asymptotic dimension in the sense of Guentner, Willet and Yu. Our results generalise those of C.~Böncicke and K.~Li to groupoids with large isotropy, including graph groupoids of directed graphs. We find that all graph C*-algebras that are stably finite have nuclear dimension at most . We also show that the nuclear dimension of the C*-algebra of a twist over has the same bound on the nuclear dimension as for and the twisted groupoid C*-algebra.

Paper Structure

This paper contains 7 sections, 32 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The quasi-orbit map
  4. Subhomogeneous C*-algebras of groupoids
  5. Nuclear dimension of C*-algebras of étale groupoids with large isotropy subgroups
  6. Application to C*-algebras of directed graphs
  7. Nuclear dimension of C*-algebras of twists

Key Result

Lemma 2.1

Suppose that $X$ and $Y$ are topological spaces with $X$ first countable. Let $f\colon X\to Y$ be an open surjection. Let $\{y_{n}\}\subset Y$ be a sequence converging to $f(x)$. Then there is a sequence $\{x_n\}\subset X$ such that $x_{n}\to x$ and such that $f(x_{n})=y_{n}$.

Theorems & Definitions (72)

  • Lemma 2.1: Sequence Lifting
  • proof
  • Lemma 2.2
  • proof
  • Example 2.3
  • Lemma 2.4
  • proof
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • ...and 62 more