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Rigidity of pseudofunction algebras of ample groupoids

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Pith's one-line read The paper proves that Hausdorff, ample groupoids are completely determined by their reduced L^p-operator algebras, their symmetrized p-pseudofunction algebras, and their I-norm completions: for p ≠ 2, any isometric isomorphism of these…

desk verdict New inverse-semigroup invariant gives clean rigidity for ample groupoids at p≠2, but the advertised p=1 case (Theorem A) is currently unsupported. read the letter →

arxiv 2506.09563 v1 pith:N53SQR4H submitted 2025-06-11 math.OA math.DSmath.GR

Eusebio Gardella , Mathias Palmstr{o}m , Hannes Thiel This is my paper · ORCID
classification math.OAmath.DSmath.GR MSC 47L1046L5520M18
keywords amplegroupoidsL^p-operatoralgebraspseudofunctionMoore-PenrosepartialisometriesinversesemigroupsrigidityI-normtight
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

An ample groupoid is a kind of generalized symmetry object built from local transformations, and the paper asks whether its convolution algebras remember every detail of it. The answer given is yes when p ≠ 2: Hausdorff, ample groupoids G and H are isomorphic exactly when their reduced L^p-operator algebras, their symmetrized p-pseudofunction algebras, or their I-norm completions are isometrically isomorphic as Banach algebras. This extends the classical group-algebra rigidity theorem from groups to groupoids. The isomorphism is not assumed to preserve the involution, and the unit space is not assumed to be known in advance; the algebra structure alone carries the information. A byproduct is a proof of a known conjecture on continuity of the Moore-Penrose inverse for L^p-operator algebras.

What carries the argument

The central object is the set of Moore-Penrose invertible partial isometries: contractive elements a for which there is a unique contractive b with a=aba, b=bab, and ab, ba hermitian. In the reduced L^p-operator algebra of a Hausdorff, étale groupoid, and for p≠2, the paper classifies these elements: each is a T-valued continuous function supported on a single compact open bisection. The proof uses the classical description of invertible isometries between L^p-spaces together with right-convolution operators that commute with the left regular representation. Homotopy classes of these elements form an inverse semigroup, and the map sending a compact open bisection to the class of its indicator function is an isomorphism onto this semigroup. A tight-groupoid reconstruction theorem turns that inverse semigroup back into the original groupoid.

What would settle it

Find a Hausdorff, ample groupoid G and p≠2 containing an element a that is contractive and has a Moore-Penrose inverse but whose natural coefficient function is supported outside a single compact open bisection, or takes values outside the unit circle T; that would break the classification on which the inverse semigroup encoding depends. Alternatively, exhibit two non-isomorphic Hausdorff, ample groupoids whose reduced L^p-operator algebras, symmetrized algebras, or I-norm completions are isometrically isomorphic; the theorem says this is impossible.

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Extended reading notes

Core claim

The central claim is a three-way rigidity statement. For every p in [1,∞)\setminus{2}, the reduced groupoid L^p-operator algebra F^p_λ(G) of a Hausdorff, ample groupoid G determines G up to topological groupoid isomorphism; the same holds for the symmetrized p-pseudofunction algebra $F^{{p,*}}$_λ(G), and at p=1 for the I-norm completion L^I(G) of the convolution algebra C_c(G). The proof works by showing that the inverse semigroup of compact open bisections of G is encoded in the algebra as the homotopy classes of Moore-Penrose invertible partial isometries. A known tight-groupoid reconstruction theorem then recovers G from that inverse semigroup. Consequently, an isometric isomorphism between two such algebras yields an isomorphism of the corresponding inverse semigroups and hence of the groupoids themselves.

Load-bearing premise

The whole reconstruction rests on classifying the contractive elements with a Moore-Penrose inverse: they must all be unimodular functions sitting on a single compact open bisection, and the symmetrized algebra must inject into the reduced one for p<2; if either fails, the algebra no longer visibly contains the semigroup from which the groupoid is rebuilt.

Editorial extensions

If this is right

  • The Banach-algebra structure of F^p_λ(G), F^{p,*}_λ(G), and L^I(G) fully determines the Hausdorff, ample groupoid G, with no involution preservation and no preselected unit-space subalgebra required.
  • At p=1, the theorem for L^I(G) extends the classical group-algebra rigidity result from groups to ample groupoids.
  • For topologically principal groupoids the result recovers earlier L^p rigidity, and for groupoids with trivial unit space it recovers the known rigidity for discrete groups.
  • The Moore-Penrose inverse is norm-continuous on convergent sequences of Moore-Penrose invertible elements exactly when their inverses are bounded, for all unital L^p-operator algebras.
  • Classifying isometric isomorphisms of these convolution algebras is equivalent to classifying the groupoids themselves.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because compact open bisections are the building blocks of the ample groupoid's topology, any invariant of the inverse semigroup B_o^c(G) becomes an invariant of the Banach algebra; this suggests K-theoretic or homological invariants of the groupoid are algebraically accessible, though the paper does not develop that.
  • The p=2 exclusion is likely essential: at p=2 these algebras are reduced C*-algebras, and non-isomorphic groups are already known to have isomorphic reduced C*-algebras, so no analogous rigidity can hold there.
  • The proof leans on total disconnectedness of the unit space through compact open bisections, so extending the theorem beyond ample groupoids would require a different way to encode the groupoid; the present method does not obviously adapt to general étale groupoids.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper studies Banach algebras associated with Hausdorff, étale groupoids: the reduced groupoid L^p-operator algebra F^p_lambda(G), its symmetrized version F^{p,*}_lambda(G), and the I-norm completion L^I(G). The main rigidity theorems (Theorems A and B) claim that, for p in [1,infinity) excluding 2, isometric isomorphisms among these algebras force topological groupoid isomorphism, and Theorem C verifies Rakočević's conjecture on continuity of the Moore-Penrose inverse in unital Banach algebras whose hermitian idempotents are ultrahermitian and commute. The proof strategy is to construct, inside each algebra, an inverse semigroup of homotopy classes of Moore-Penrose partial isometries, identify it with the inverse semigroup B^o_c(G) of compact open bisections, and then invoke Exel's tight groupoid reconstruction. Section 2 develops the general semigroup machinery, Section 3 identifies S_pi(F^p_lambda(G)) with B^o_c(G), and Section 4 applies this to the symmetrized algebras and to L^I(G), taking L^I(G)=F^{1,*}_lambda(G).

Significance. If correct, the results substantially extend Wendel's theorem from discrete groups to Hausdorff, ample groupoids, and they improve on earlier rigidity results by not requiring the isomorphism to preserve the unit-space subalgebra or the involution. The introduction of Moore-Penrose partial isometries and their homotopy classes as an inverse-semigroup invariant is original and likely to be influential. The central support argument in Proposition 3.11 is detailed and convincing, and the verification of Rakočević's conjecture for L^p-operator algebras is a valuable byproduct. The derivation is not circular: S_pi(F^p_lambda(G)) is shown to be isomorphic to the external semigroup B^o_c(G), and Exel's reconstruction then recovers G. The main weakness is that the p=1 case of the symmetrized reconstruction is not proved: the argument for Proposition 4.6 uses an isometric diagonal embedding that is unavailable at p=1, and Lemma 4.5 asserts the needed injectivity at p=1 without proof.

major comments (4)
  1. [§4, Proposition 4.6 and Theorem 4.7] The p=1 case is not proved. The first assertion of Proposition 4.6 is justified by applying Lemma 2.21 through the isometric diagonal map F^{p,*}_lambda(G) -> F^p_lambda(G) ⊕ F^q_lambda(G) described in §4.4. For p=1 the dual exponent is q=∞, and F^∞_lambda(G) is not defined (Definition 3.3 treats only p in [1,∞)). Moreover, Lemma 2.21 at p=1 requires the algebra to admit a nondegenerate isometric representation on an L^1-space, while the introduction explicitly states that L^I(G) is not representable on an L^p-space. Thus the proof gives no information about hermitian idempotents in L^I(G)=F^{1,*}_lambda(G). Since Theorem 4.7, Corollary 4.8, and Theorem A all specialize to p=1, the headline rigidity theorem is unsupported at p=1.
  2. [§4, Lemma 4.5] The p=1 case of injectivity of the canonical map φ_1: L^I(G) -> F^1_lambda(G) is asserted in a single sentence, but it is not formal. The map is contractive but not isometric: for example, in a groupoid with two arrows sharing the same source and having distinct ranges, the indicator function of their union has I-norm 2 but F^1_lambda-norm 1. Therefore injectivity of the map on C_c(G) does not automatically extend to the completed map. Even if injectivity were established, Lemma 2.20(2) requires the map to be isometric in order to pull back hermitianness or ultrahermitianness, so a merely contractive injective map cannot transfer the structural conclusions of Lemma 2.21 back to L^I(G). A separate argument for p=1 is needed.
  3. [§4, Proposition 4.6, final sentence] The injectivity of the induced map S_pi(F^{p,*}_lambda(G)) -> S_pi(F^p_lambda(G)) is not proved; the text says 'We omit the details.' This is load-bearing because Theorem 4.7 requires an isomorphism of inverse semigroups. The cited 'arguments as in the proof of Lemma 3.12' need to be written out, including why a path that is continuous in F^p_lambda(G) lies in PIMP(F^{p,*}_lambda(G)) and is continuous in the F^{p,*}-norm. This is likely fixable for p in (1,2), but it is currently a gap.
  4. [§4, Proposition 4.6, surjectivity of Ψ_p] Surjectivity of Ψ_p is said to be 'immediate' because MP-partial isometries in F^p_lambda(G) are supported on compact open bisections and lie in C_c(G). This requires verification in F^{p,*}_lambda(G), not only in F^p_lambda(G): one must check that f^*f=1_{s(B)} and f f^*=1_{r(B)} are hermitian idempotents in F^{p,*}_lambda(G) and that the relevant norms are contractive. Since φ_p is not isometric, these facts cannot be quoted from Lemma 3.8 or Lemma 3.7 without additional argument. A short direct verification should be included.
minor comments (5)
  1. [§3, Lemma 3.12] There are typos: 'homotomopy' should be 'homotopy' and 'funciton' should be 'function'.
  2. [§2, Theorem 2.17] In the proof, 'we need to shown that' should be 'we need to show that'.
  3. [§2, internal labels] The displayed statements labelled 'Theorem 2.14' and 'Theorem 2.23' are referred to elsewhere as 'Lemma 2.14' and 'Lemma 2.23'; the numbering and naming should be made consistent.
  4. [§2, Definition 2.27] The notation for the homotopy-class inverse semigroup is typeset inconsistently as S_pi(A) and S_pi(A) in different places; one symbol should be used throughout.
  5. [§1 and §2.19] The phrase 'not representable on an L^p-space' is ambiguous: it should say whether L^I(G) is not representable on any L^p-space or merely that a particular representation is not isometric. This matters because Lemma 2.21 has a p=1 hypothesis formulated in terms of such representations.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular step found: the rigidity chain S_pi(A) ≅ B^o_c(G) → G_tight is computed from external L^p geometry (Lamperti) and Exel's reconstruction, not from the target isomorphism; the p=1 branch (Lemma 4.5, Proposition 4.6) is a proof gap, not circularity.

full rationale

The central chain is not circular: S_pi(A) is defined purely analytically (Definition 2.27) from contractive elements admitting contractive Moore-Penrose inverses with hermitian projections, and Theorem 3.13 proves S_pi(F^p_λ(G)) ≅ B^o_c(G) by a genuine computation — Lemma 3.11 (via Lamperti's theorem and [CGT24, Prop. 2.7]) shows MP-partial isometries are T-valued on compact open bisections, and Lemma 3.12 collapses the T-valued functions to indicators by a homotopy argument — after which Exel's external reconstruction [Exe10, Thm. 4.8] yields G ≅ G_tight(S_pi(F^p_λ(G))); no equation in this chain is equivalent to the target isomorphism by construction, and no fitted parameters appear. The self-citations [CGT24], [GT22], and [Gar21] carry the heavy analytic inputs, but they are published, parameter-free theorems whose assumptions do not include groupoid rigidity, so they qualify as independent evidence and do not make the argument circular. The flagged weakness is the p=1 branch: Lemma 4.5 asserts the injectivity of φ_1: L^I(G) → F^1_λ(G) ('which is injective') without proof, and Proposition 4.6 derives the required ultrahermitian/commuting properties from the isometric diagonal embedding F^{p,*}_λ(G) → F^p_λ(G) ⊕ F^q_λ(G), which exists only for p ∈ (1,∞) since F^∞_λ(G) is never defined (Definition 3.3) and Lemma 2.21's p=1 hypothesis (a nondegenerate isometric L^1-representation) fails for L^I(G) by the paper's own admission that L^I(G) is not representable on an L^p-space; consequently Theorem 4.7 at p=1, hence Corollary 4.8, rest on an unproved assertion, and the homotopy-injectivity details are 'omitted'. This is a missing-proof correctness risk, not a self-definitional reduction, so the circularity score stays at 1.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on external theorems such as Exel reconstruction, Lamperti's theorem, Vidav-Palmer, and the CGT24 description of hermitian elements on Lp-spaces. There are no fitted constants and no newly postulated physical or mathematical entities beyond the directly defined inverse semigroup S_pi(A), which is constructed from A rather than assumed.

assumptions (5)
  • domain assumption Exel's reconstruction theorem: for an ample groupoid G, G is isomorphic to the tight groupoid G_tight(B_c^o(G)) (Exe10, Theorems 3.6 and 4.8).
    Invoked in Lemma 4.1 to convert an isomorphism of inverse semigroups into an isomorphism of groupoids. External result.
  • domain assumption Lamperti's description of invertible isometries between Lp-spaces: such an isometry is a weighted composition operator (Gar21, Theorem 2.12).
    Used in Proposition 3.11 to conclude lambda(a) is a weighted composition operator and therefore has support forming a bisection.
  • domain assumption Hermitian idempotents on Lp-spaces are multiplication operators by characteristic functions, and hermitian operators on Lp (p not equal to 2) commute (CGT24, Propositions 2.6, 2.7 and Corollary 2.8).
    Core input for Lemma 2.16 and Proposition 2.21, which imply PI_MP forms an inverse semigroup.
  • standard math Vidav-Palmer theorem: a unital Banach algebra with A = A_h + i A_h is a C*-algebra.
    Used in Proposition 2.5 and Lemma 2.23 to identify the core and the projection algebra D with a commutative C*-algebra C(X).
  • domain assumption Rakocevic's conjecture holds for C*-algebras (Rak93, Theorem 2.2).
    Used in Lemma 2.25 to handle the p=2 case for L2-operator algebras.

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Pith. "Pith review of Rigidity of pseudofunction algebras of ample groupoids." pith.science (2026). https://pith.science/paper/N53SQR4H

@misc{pith2026250609563,
  author       = {Pith},
  title        = {Pith review of: Rigidity of pseudofunction algebras of ample groupoids},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N53SQR4H}},
  note         = {Machine review of arXiv:2506.09563}
}
abstract

We show that a Hausdorff, ample groupoid $\mathcal{G}$ can be completely recovered from the $I$-norm completion of $C_c(\mathcal{G})$. More generally, we show that this is also the case for the algebra of symmetrized $p$-pseudofunctions, as well as for the reduced groupoid $L^p$-operator algebra, for $p\neq 2$. Our proofs are based on a new construction of an inverse semigroup built from Moore-Penrose invertible partial isometries in an $L^p$-operator algebra. Along the way, we verify a conjecture of Rako\v{c}evi\'{c} concerning the continuity of the Moore-Penrose inverse for $L^p$-operator algebras.

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