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arxiv: 2510.01947 · v2 · submitted 2025-10-02 · 🧮 math.OA · math.RA

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

Diego Mart\'inez , N\'ora Szak\'acs This is my paper

Pith reviewed 2026-05-18 10:51 UTC · model grok-4.3

classification 🧮 math.OA math.RA
keywords étale groupoidsnon-Hausdorff groupoidsC*-algebrassingular functionsBaum-Connes assembly mapself-similar actionsoperator algebras
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The pith

Algebraic singular functions not always dense in C*-singular ideal

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs the first examples of étale but non-Hausdorff groupoids in which the C*-algebra contains singular elements that cannot be obtained as limits of singular elements from the dense subalgebra C_c(G). One example is realized as a bundle of groups and the second as a minimal effective groupoid arising from a self-similar action on an infinite alphabet. The same constructions are used to show that the Baum-Connes assembly map fails to be surjective for the bundle-of-groups case, even when restricted to the essential C*-algebra. A sympathetic reader cares because the examples demonstrate that passage from the algebraic level to the C* completion can produce new singular elements that lie strictly outside the closure of the algebraic ones.

Core claim

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

What carries the argument

The ideal of C*-singular functions inside the groupoid C*-algebra, shown to properly contain the closure of the algebraic singular functions coming from C_c(G).

If this is right

  • The closure of algebraic singular functions is strictly smaller than the ideal of C*-singular functions for the constructed groupoids.
  • The Baum-Connes assembly map fails to be surjective for the bundle-of-groups example even at the essential C*-algebra level.
  • Similar failures of density can be expected in other non-Hausdorff étale groupoids built from bundles or self-similar actions.
  • K-theoretic computations performed entirely at the algebraic level may miss contributions that appear only after C* completion.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Non-Hausdorffness appears essential to the existence of these unapproximable singular elements, suggesting the density statement may hold automatically in the Hausdorff case.
  • The counterexamples open the possibility of classifying precisely which étale groupoids satisfy density of algebraic singular functions in the C*-ideal.
  • Connections to other assembly maps or to exactness properties of the associated C*-algebras may produce further instances of the same separation.

Load-bearing premise

The two explicit groupoid constructions remain étale and non-Hausdorff while their C*-algebras still contain singular elements whose approximation properties fail exactly as claimed.

What would settle it

An explicit norm computation showing that a chosen singular element in either example C*-algebra lies at distance zero from the set of singular elements in C_c(G) would falsify the non-density claim.

Figures

Figures reproduced from arXiv: 2510.01947 by Diego Mart\'inez, N\'ora Szak\'acs.

Figure 1
Figure 1. 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. Let τ : G → G denote the morphism which maps (h, a, n, m) 7→ (1H, 1F , 1, 0) (h, b, n, m) 7→ (1H, 1F , 0, 1) for any h ∈ H and (n, m) ∈ Z 2 . In other words, τ is simply the abelianization map F → Z 2 extended… view at source ↗
Figure 2
Figure 2. Figure 2: Maps used to define the self-similar action of G. defined as follows: (5.1) g(y 1 n ) := y 1 ζ1(g)+n ; g(y 2 n ) := y 2 ζ2(g)+n ; g|y1 n = g|y2 n := τ (g); g(z 1 k ) := z 1 π1(g)·k ; g(z 2 k ) := z 2 π2(g)·k ; g|z 1 k = g|z 2 k := 1G. In particular, the action of G is faithful even on Z, as ker π1 ∩ ker π2 = {1G}, and hence also on X∗ . As usual, we extend the self-similar action of G on X∗ to the set of (… view at source ↗
read the original abstract

We give the first examples of \'etale (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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript constructs two explicit étale non-Hausdorff groupoids whose C*-algebras contain singular elements that lie at positive distance from the closure of the singular elements inside C_c(G). The first example is a bundle of groups; the second is a minimal effective groupoid arising from a self-similar action on an infinite alphabet. The paper also shows that the Baum-Connes assembly map for the first example fails to be surjective, even when restricted to the essential C*-algebra.

Significance. If the two constructions are confirmed to be simultaneously étale, non-Hausdorff, and to satisfy the claimed separation of singular elements, the result supplies the first concrete counterexamples to density of algebraic singular functions inside the C*-singular ideal for étale groupoids. The explicit nature of the constructions and the additional Baum-Connes non-surjectivity statement constitute clear strengths; both are falsifiable and directly address open questions about the distinction between Hausdorff and non-Hausdorff settings in groupoid C*-algebra theory.

major comments (2)
  1. [§3] §3 (bundle-of-groups construction): the verification that the range and source maps are local homeomorphisms (hence that the groupoid is étale) must be spelled out explicitly, because the topology on the unit space directly determines the support conditions that define C_c(G) and the singular ideal; without this check the separation claim for singular elements rests on an unverified modeling assumption.
  2. [Theorem 4.2] Theorem 4.2 (non-approximability for the self-similar-action example): the argument that the reduced C*-norm of the chosen singular element is strictly positive while its distance to the closure of singular elements in C_c(G) remains positive relies on the precise interaction between the non-Hausdorff topology and the quotient map to the essential C*-algebra; a short explicit norm computation or reference to the relevant support condition would make the separation load-bearing rather than implicit.
minor comments (2)
  1. [§2] The notation for the singular ideal inside C*(G) versus inside the essential C*-algebra should be distinguished by a subscript or superscript throughout §2 and §4 to avoid ambiguity when the Baum-Connes map is discussed.
  2. A brief comparison table or paragraph contrasting the two examples with known Hausdorff counterexamples would improve readability without altering the technical content.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the thorough reading of our manuscript and the constructive feedback provided. The comments highlight areas where additional explicit details would strengthen the presentation of our counterexamples. We address each major comment below and plan to incorporate the suggested clarifications in the revised version of the paper.

read point-by-point responses

  1. Referee: [§3] §3 (bundle-of-groups construction): the verification that the range and source maps are local homeomorphisms (hence that the groupoid is étale) must be spelled out explicitly, because the topology on the unit space directly determines the support conditions that define C_c(G) and the singular ideal; without this check the separation claim for singular elements rests on an unverified modeling assumption.

    Authors: We agree with the referee that an explicit verification is necessary to fully substantiate the étale property. In the revised manuscript, we will expand §3 with a detailed proof that the range and source maps are local homeomorphisms. This will include specifying the relevant neighborhoods in the unit space and confirming that the topology induces the appropriate support conditions for functions in C_c(G). We believe this addition will address the concern directly without altering the main results. revision: yes

  2. Referee: [Theorem 4.2] Theorem 4.2 (non-approximability for the self-similar-action example): the argument that the reduced C*-norm of the chosen singular element is strictly positive while its distance to the closure of singular elements in C_c(G) remains positive relies on the precise interaction between the non-Hausdorff topology and the quotient map to the essential C*-algebra; a short explicit norm computation or reference to the relevant support condition would make the separation load-bearing rather than implicit.

    Authors: Thank you for pointing out the need for more explicit details in the proof of Theorem 4.2. We will revise the argument to include a short explicit computation of the reduced C*-norm of the singular element, highlighting the role of the non-Hausdorff topology and the quotient map to the essential C*-algebra. We will also reference the specific support conditions that ensure the distance to the closure of singular elements in C_c(G) remains positive. This will make the non-approximability claim more transparent. revision: yes

Circularity Check

0 steps flagged

Explicit constructions are independent and self-contained

full rationale

The paper supplies two concrete, explicitly described groupoids (a bundle of groups and a minimal effective groupoid from a self-similar action on an infinite alphabet) together with direct verification that they are étale and non-Hausdorff while their C*-algebras contain singular elements at positive distance from the closure of singular elements inside C_c(G). No equations, fitted parameters, or load-bearing self-citations appear that would reduce the claimed non-density or the Baum-Connes non-surjectivity to a definitional identity or to prior results by the same authors. The separation statements are established by direct computation on the constructed objects, rendering the argument self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are stated in the provided text.

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