[Paper Review] Non-Hausdorff groupoids

Ruy Exel|ArXiv.org|Dec 22, 2008

Advanced Operator Algebra Research9 references12 citations

TL;DR

This paper constructs explicit examples of non-Hausdorff, étale, essentially principal groupoids where three fundamental results—(A) maximality of the continuous functions on the unit space in the reduced groupoid C*-algebra, (B) every nonzero ideal intersecting the unitary subalgebra, and (C) normalizers having bissected supports—fail. The failure occurs due to non-Hausdorff topology at the origin of a union of two lines under a Z₂-action, demonstrating that these results are inherently dependent on the Hausdorff assumption.

ABSTRACT

We present examples of non-Hausdorff, etale, essentially principal groupoids for which three results, known to hold in the Hausdorff case, fail. These results are: (A) the subalgebra of continuous functions on the unit space is maximal abelian within the reduced groupoid C*-algebra, (B) every nonzero ideal of the reduced groupoid C*-algebra has a nonzero intersection with the subalgebra of continuous functions on the unit space, and (C) the open support of a normalizer is a bissection.

Motivation & Objective

• To demonstrate that three key results in groupoid C*-algebra theory—maximal abelian subalgebras, ideal intersection with unit space, and normalizer support as bissections—fail in the non-Hausdorff setting.
• To construct explicit counterexamples using the groupoid of germs for a Z₂-action on a union of two real lines, with non-Hausdorff topology at the origin.
• To clarify that these results, widely used in Cuntz-Krieger and graph C*-algebras, depend crucially on the Hausdorff assumption.
• To provide a precise analysis of the failure of the conditional expectation argument in non-Hausdorff cases, particularly regarding the norm control of functions on the unit space.
• To establish that the standard uniqueness and faithfulness theorems in the reduced groupoid C*-algebra fail without Hausdorffness, even under essential principality.

Proposed method

• Construct a non-Hausdorff étale groupoid G as the groupoid of germs for the Z₂-action on Z = [-1,1]×{0} ∪ {0}×[-1,1] generated by reflections σₓ and σᵧ.
• Define the unit space G⁽⁰⁾ as Z, and identify the isotropy group at the origin as nontrivial, leading to non-Hausdorff behavior.
• Define a function f ∈ C_c(G) as f = f₁ − fₓ − fᵧ + fₓᵧ, where fₐ is the characteristic function of the bisection Uₐ, and show f(0) = 1 while f vanishes on all other units.
• Use the GNS representation associated with a Dirac state at the origin to show that ||π(f)|| = 1, but supₓ∈G⁽⁰⁾ |f(x)| = 1, yet f is not in the closure of C₀(G⁽⁰⁾) in the C*-norm.
• Prove that the conditional expectation E: C_r^*(G) → C₀(G⁽⁰⁾) fails to satisfy ||E(f)|| ≤ ||π(f)|| in the non-Hausdorff case, breaking the standard faithfulness argument.
• Use a net of approximate units (ξᵥ) supported near the origin to show that |⟨π(f)ξᵥ, ξᵥ⟩| → |f(x)| for x ∈ G⁽⁰⁾, but the norm control fails due to non-Hausdorff topology.

Experimental results

Research questions

• RQ1 Does the subalgebra C₀(G⁽⁰⁾) remain maximal abelian in C_r^*(G) when G is non-Hausdorff and étale?
• RQ2 Can a nonzero ideal in C_r^*(G) avoid intersecting C₀(G⁽⁰⁾) when G is non-Hausdorff and essentially principal?
• RQ3 Is the open support of a normalizer in C_r^*(G) necessarily a bissection in the non-Hausdorff case?
• RQ4 Can the standard proof of faithfulness of representations on C₀(G⁽⁰⁾) be extended to non-Hausdorff groupoids?
• RQ5 What role does the Hausdorff condition play in the validity of uniqueness theorems for groupoid C*-algebras?

Key findings

• The function f = f₁ − fₓ − fᵧ + fₓᵧ satisfies f(0) = 1 and f(γ) = 0 for all other γ ∈ G⁽⁰⁾, yet ||π(f)|| = 1, showing that supₓ∈G⁽⁰⁾ |f(x)| = 1 is not bounded by ||π(f)|| in the non-Hausdorff case.
• The conditional expectation E: C_r^*(G) → C₀(G⁽⁰⁾) fails to satisfy ||E(f)|| ≤ ||π(f)|| for this f, invalidating the standard argument for faithfulness.
• The representation π of C_r^*(G) is faithful on C₀(G⁽⁰⁾), but π(f) has norm 1 while f is not in the closure of C₀(G⁽⁰⁾), showing C₀(G⁽⁰⁾) is not maximal abelian.
• The ideal J = ker(π) is nonzero but satisfies J ∩ C₀(G⁽⁰⁾) = 0, violating property (B), even though G is essentially principal.
• The normalizer u corresponding to f has open support equal to G, which is not a bissection, violating property (C).
• The failure of the norm control argument stems from the non-Hausdorff topology at the origin, where the net of approximate units does not converge in the strong operator topology to a limit that preserves pointwise bounds.