arxiv: 2604.04663 · v1 · submitted 2026-04-06 · 🧮 math.OA

Recognition: 2 theorem links

· Lean Theorem

On the Haagerup property for partial crossed products

Chaitanya J. Kulkarni, Md Amir Hossain

Pith reviewed 2026-05-10 19:27 UTC · model grok-4.3

classification 🧮 math.OA

keywords Haagerup propertypartial crossed productsC*-algebraspartial actionsdiscrete groupsinductive limitsoperator algebras

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The pith

The Haagerup property holds for a reduced partial crossed product if and only if it holds for both the C*-algebra and the partial action.

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

The paper introduces a definition of the Haagerup property for partial actions of discrete groups on C*-algebras. It proves that the reduced partial crossed product has this property if and only if the C*-algebra A and the partial action α both have it. Consequently, the property for the crossed product is equivalent to A and the group G both having the Haagerup property. The authors also establish that the Haagerup property is preserved under inductive limits and use this to investigate inductive limits of partial crossed products.

Core claim

We introduce the Haagerup property for partial actions of discrete groups on C*-algebras. We prove that the partial crossed product A⋊_{α,r} G has the Haagerup property if and only if both A and the partial action α have the Haagerup property. As a consequence, we obtain an equivalence between the Haagerup property of the partial crossed product and that of the underlying C*-algebra and the acting group. We also show that the Haagerup property is preserved under inductive limits and apply this result to study the Haagerup property of inductive limits of partial crossed products.

What carries the argument

The definition of the Haagerup property for partial actions, which enables the if-and-only-if characterization for the reduced partial crossed product A ⋊_{α,r} G.

Load-bearing premise

The introduced definition of the Haagerup property for partial actions must be compatible with the reduced crossed product construction and agree with the standard definition when the action is global.

What would settle it

An explicit partial dynamical system (A, G, α) where the reduced crossed product A ⋊_{α,r} G has the Haagerup property but A or α does not, or where A and α have it but the crossed product does not.

read the original abstract

Let $(A,G,\alpha)$ be a partial dynamical system and let $A\rtimes_{\alpha,r} G$ denote the associated reduced partial crossed product. In this article, we introduce the Haagerup property for partial actions of discrete groups on $C^*$-algebras. We prove that the partial crossed product $A\rtimes_{\alpha,r} G$ has the Haagerup property if and only if both $A$ and the partial action $\alpha$ have the Haagerup property. As a consequence, we obtain an equivalence between the Haagerup property of the partial crossed product and that of the underlying $C^*$-algebra and the acting group. We also show that the Haagerup property is preserved under inductive limits and apply this result to study the Haagerup property of inductive limits of partial crossed products.

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

0 major / 1 minor

Summary. The paper introduces a definition of the Haagerup property for partial actions of discrete groups on C*-algebras. It proves that the reduced partial crossed product A ⋊_{α,r} G has the Haagerup property if and only if both A and the partial action α have the Haagerup property. As a consequence, this is equivalent to the Haagerup property of the underlying C*-algebra A and the acting group G. The paper also shows preservation of the Haagerup property under inductive limits and applies the results to inductive limits of partial crossed products.

Significance. If the result holds, the work provides a natural extension of the Haagerup property to partial dynamical systems and crossed products, yielding a clean characterization that facilitates the construction and study of C*-algebras with this approximation property. The new definition for partial actions, which is crafted to agree with the classical definition on global actions, together with the equivalence and inductive-limit preservation results, constitutes a useful addition to the literature on operator-algebraic approximation properties.

minor comments (1)

[Abstract and Introduction] The abstract states the consequence relating the Haagerup property of the partial crossed product to that of A and G, but the manuscript should make explicit in the introduction or main theorem statement how the Haagerup property of the partial action α is shown to be equivalent to that of G.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The referee's summary correctly identifies the core results: the introduction of the Haagerup property for partial actions, the equivalence for the reduced partial crossed product, the consequence for the underlying algebra and group, and the preservation under inductive limits.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper introduces an independent definition of the Haagerup property for partial actions of discrete groups on C*-algebras, explicitly required to agree with the classical definition on global actions and to be compatible with the reduced crossed-product construction. The central if-and-only-if theorem is then proved in both directions by exhibiting explicit approximating sequences (or affine actions) that descend from the partial action to the crossed product and lift in the converse direction. Because the new definition is stated first as a standalone extension and the equivalence is derived from it rather than built into the definition itself, no step reduces by construction to a fitted parameter, self-referential clause, or load-bearing self-citation. The derivation chain therefore remains self-contained once the definition is accepted.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The claim rests on the standard theory of partial dynamical systems and reduced crossed products in C*-algebras, plus the new definition of the Haagerup property for partial actions. No numerical parameters are fitted.

axioms (2)

domain assumption Standard definitions and constructions of partial dynamical systems (A, G, α) and the reduced partial crossed product A ⋊_{α,r} G.
The paper assumes these are known from the literature on partial actions.
domain assumption The Haagerup property for C*-algebras and for groups is the standard one from prior work.
The new partial-action version is built on top of these.

invented entities (1)

Haagerup property for partial actions no independent evidence
purpose: To capture approximation properties of partial dynamical systems in a manner compatible with crossed-product constructions.
This is a new definition introduced by the authors.

pith-pipeline@v0.9.0 · 5443 in / 1380 out tokens · 60718 ms · 2026-05-10T19:27:45.976817+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
We introduce the Haagerup property for partial actions... prove that the partial crossed product A⋊α,rG has the Haagerup property if and only if both A and the partial action α have the Haagerup property (Theorem 3.7).
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
Definition 3.1. ... positive definite functions {hn:G→Z(A)}⊆C0,τ(G,A) such that hn(g)∈Dg ... hn→1 pointwise w.r.t. τ.

Reference graph

Works this paper leans on

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