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Algebraic singular functions are not always dense in the ideal of $C^*$-singular functions

Diego Martínez, Nóra Szakács

TL;DR

The paper constructs two explicit étale, non-Hausdorff groupoids whose reduced C*-algebras contain singular elements not approximable by algebraic singular functions, thereby showing that the $C^*$-singular ideal $J$ is not densely generated by $\mathcal{C}_c(\mathcal{G})$-elements. The first example is a bundle of groups for which the essential Baum–Connes assembly map fails to be surjective, even at the level of the essential algebra, illustrating pathological K-theoretic behavior. The second example, based on a self-similar action on an infinite alphabet, yields a minimal and effective groupoid with the same non-density phenomenon, reinforcing that non-Hausdorff dynamics can produce robust singular structures. Together these results answer negatively longstanding questions about the density of algebraic singular functions and highlight intricate interactions between non-Hausdorff dynamics, singular ideals, and Baum–Connes-type phenomena in groupoid C*-algebras.

Abstract

We give the first examples of étale (non-Hausdorff) groupoids $\mathcal G$ whose $C^*$-algebras contain singular elements that cannot be approximated by singular elements in $\mathcal C_c(\mathcal G)$. We provide two examples: one is a bundle of groups, and the other a minimal and effective groupoid constructed from a self-similar action on an infinite alphabet. Moreover, we also prove that the Baum--Connes assembly map for the first example is not surjective, not even on the level of its essential $C^*$-algebra.

Algebraic singular functions are not always dense in the ideal of $C^*$-singular functions

TL;DR

The paper constructs two explicit étale, non-Hausdorff groupoids whose reduced C*-algebras contain singular elements not approximable by algebraic singular functions, thereby showing that the -singular ideal is not densely generated by -elements. The first example is a bundle of groups for which the essential Baum–Connes assembly map fails to be surjective, even at the level of the essential algebra, illustrating pathological K-theoretic behavior. The second example, based on a self-similar action on an infinite alphabet, yields a minimal and effective groupoid with the same non-density phenomenon, reinforcing that non-Hausdorff dynamics can produce robust singular structures. Together these results answer negatively longstanding questions about the density of algebraic singular functions and highlight intricate interactions between non-Hausdorff dynamics, singular ideals, and Baum–Connes-type phenomena in groupoid C*-algebras.

Abstract

We give the first examples of étale (non-Hausdorff) groupoids whose -algebras contain singular elements that cannot be approximated by singular elements in . We provide two examples: one is a bundle of groups, and the other a minimal and effective groupoid constructed from a self-similar action on an infinite alphabet. Moreover, we also prove that the Baum--Connes assembly map for the first example is not surjective, not even on the level of its essential -algebra.

Paper Structure

This paper contains 17 sections, 22 theorems, 73 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ample groupoids
  4. Groupoids associated to self-similar groups
  5. Algebras associated to groupoids
  6. Spectral radii in non-amenable groups
  7. The strategy
  8. First example: a bundle of groups
  9. Description of the groupoid
  10. Scattering
  11. Finding a B-singular function.
  12. Showing a*b is not in Jalg.
  13. The groupoid G is a counterexample to the (essential) Baum--Connes Conjecture
  14. Second example: a minimal and effective groupoid
  15. Scattering.
  16. ...and 2 more sections

Key Result

Proposition 2.2

[proposition]prop:stein description If $S$ is an inverse semigroup with zero and $\mathcal{G}$ its universal groupoid, then $\mathbb{C}(\mathcal{G}) = \operatorname{span} \{\chi_{(s,D(s^*s))} \colon s \in S\setminus \{0\}\}$.

Figures (2)

  • Figure 1: The self-similar action of $G$, depicting how different elements of $G$ act on letters (viewed as the first level of the tree associated to finite words in $X$), and what their sections are at each letter.
  • Figure 2: Maps used to define the self-similar action of $G$.

Theorems & Definitions (38)

  • Proposition 2.2: see BenSteinAlgPaper*Theorem 5.3
  • Proposition 2.3: see ExelBigPaper*Proposition 3.10
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • Remark 3.1
  • Theorem 4.1
  • Lemma 4.2
  • ...and 28 more