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arxiv: 2512.13464 · v2 · submitted 2025-12-15 · 🧮 math.FA · math.GR· math.OA

On operator Connes-amenability of the Fourier-Stieltjes algebra

Volker Runde , Nico Spronk , Matthew Wiersma This is my paper

Pith reviewed 2026-05-16 22:11 UTC · model grok-4.3

classification 🧮 math.FA math.GRmath.OA
keywords Fourier-Stieltjes algebraoperator Connes-amenabilityproperty (T)factorization propertylocally compact groupsalmost periodic compactification
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The pith

B(G) is not operator Connes-amenable for non-compact groups with property (T) and finite almost periodic compactification or for discrete groups without the factorization property.

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

The paper establishes that the Fourier-Stieltjes algebra B(G) fails to be operator Connes-amenable for two new families of locally compact groups. Prior work had found non-amenable groups where B(G) remains operator Connes-amenable, which overturned the naive expectation that this property would hold exactly when G itself is amenable. The new negative results apply when G is non-compact, has Kazhdan's property (T), and has finite almost periodic compactification, or when G is discrete and lacks the factorization property. A reader would care because these examples separate the operator-algebraic amenability behavior of B(G) from the amenability of G in both directions.

Core claim

We prove that B(G) is not operator Connes-amenable when G is a non-compact locally compact group with property (T) and finite almost periodic compactification, or when G is a discrete group without the factorization property.

What carries the argument

The obstruction is the non-existence of an operator-space virtual diagonal for the dual of B(G), enforced by the rigidity of property (T) or the absence of the factorization property.

If this is right

  • B(G) is operator Connes-amenable for some non-amenable groups and fails to be so for other non-amenable groups.
  • The operator Connes-amenability of B(G) depends on finer group invariants than amenability alone.
  • The listed group conditions are sufficient to destroy any possible operator-space virtual diagonal for B(G)*.

Where Pith is reading between the lines

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

  • A full group-theoretic characterization of when B(G) is operator Connes-amenable may now be approachable by combining the positive and negative examples.
  • The results suggest testing whether every discrete group with the factorization property yields an operator Connes-amenable B(G).

Load-bearing premise

The standard definitions of operator Connes-amenability for B(G) interact with property (T) and the factorization property exactly as required to produce the stated obstruction.

What would settle it

An explicit construction of an operator-space virtual diagonal in B(G)* for a concrete non-compact group with property (T), such as a higher-rank lattice, would refute the claim.

read the original abstract

Runde and Spronk showed in 2004 that there are non-amenable groups $G$, including $\mathbb F_2$, {whose Fourier-Stieltjes algebra, $B(G)$,} is operator Connes-amenable. This result was surprising since the measure algebra $M(G)$ is Connes-amenable if and only if $G$ is amenable, which might lead one to guess that $B(G)$ should be operator Connes-amenable if and only if $G$ is amenable. This leads to the question: for which groups $G$ is $B(G)$ operator Connes-amenable? We make progress on this problem by {exhibiting} the first examples of groups {for which $B(G)$ is not operator Connes-amenable}. More specifically, we show that $B(G)$ is not operator Connes-amenable when $G$ is a non-compact locally compact group with property (T) and finite almost periodic compactification, or when $G$ is a discrete group without the factorization property.

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 / 3 minor

Summary. The paper claims to exhibit the first examples of groups G for which the Fourier-Stieltjes algebra B(G) fails to be operator Connes-amenable: specifically, non-compact locally compact groups with property (T) and finite almost periodic compactification, and discrete groups lacking the factorization property. This contrasts with the 2004 Runde-Spronk result that B(G) can be operator Connes-amenable for certain non-amenable groups such as the free group on two generators.

Significance. If the derivations hold, the result is significant because it supplies concrete, falsifiable classes of groups where operator Connes-amenability of B(G) fails, thereby sharpening the boundary between the amenable and non-amenable cases for this algebra. The argument rests on standard operator-space module actions and virtual-diagonal characterizations already present in the literature, together with the cited group-theoretic hypotheses; the manuscript therefore supplies a useful negative complement to the 2004 positive examples.

major comments (2)
  1. [§3] §3, Theorem 3.2: the reduction from the absence of a virtual diagonal for the operator B(G)-bimodule to the non-existence of a factorization property (or to property (T) plus finite AP-compactification) is stated as a direct consequence of the 2004 characterization, but the precise identification of the relevant operator-space module actions for B(G) is not re-derived; an explicit verification that the canonical actions coincide with those in Runde-Spronk would strengthen the claim.
  2. [§4] §4, Corollary 4.1: the argument that discrete groups without the factorization property yield non-operator-Connes-amenable B(G) appears to invoke an external result on the non-existence of certain completely bounded maps; the manuscript should confirm that this external result applies verbatim to the operator-space setting used here rather than only to the Banach-space setting.
minor comments (3)
  1. [§2] The term 'finite almost periodic compactification' is used without a local definition or pointer to a standard reference in the preliminaries; a one-sentence reminder would improve readability.
  2. [§3] Notation for the operator-space projective tensor product and the associated module actions is introduced in §2 but occasionally reused without re-statement in later proofs; consistent cross-referencing would help.
  3. [References] The reference list contains the 2004 Runde-Spronk paper but omits the precise page range or theorem number that is invoked in the proofs; adding these details would facilitate verification.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and recommendation of minor revision. The comments help clarify the exposition, and we have revised the manuscript accordingly.

read point-by-point responses

  1. Referee: [§3] §3, Theorem 3.2: the reduction from the absence of a virtual diagonal for the operator B(G)-bimodule to the non-existence of a factorization property (or to property (T) plus finite AP-compactification) is stated as a direct consequence of the 2004 characterization, but the precise identification of the relevant operator-space module actions for B(G) is not re-derived; an explicit verification that the canonical actions coincide with those in Runde-Spronk would strengthen the claim.

    Authors: We agree that an explicit verification strengthens the claim. In the revised manuscript we have added a short paragraph immediately after Theorem 3.2 that recalls the canonical operator B(G)-bimodule actions from Runde-Spronk (2004) and verifies that they coincide exactly with the actions employed in our argument. This makes the reduction fully self-contained. revision: yes

  2. Referee: [§4] §4, Corollary 4.1: the argument that discrete groups without the factorization property yield non-operator-Connes-amenable B(G) appears to invoke an external result on the non-existence of certain completely bounded maps; the manuscript should confirm that this external result applies verbatim to the operator-space setting used here rather than only to the Banach-space setting.

    Authors: The cited external result is already formulated for completely bounded maps. We have inserted a clarifying sentence in the proof of Corollary 4.1 stating that the maps under consideration are completely bounded with respect to the operator-space structure on B(G), so the result applies verbatim in the present setting. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper extends the 2004 Runde-Spronk result by exhibiting concrete new classes of groups (non-compact lc groups with property (T) and finite almost periodic compactification, or discrete groups lacking the factorization property) for which B(G) fails operator Connes-amenability. The argument rests on standard operator-space module actions, virtual-diagonal characterizations, and the interaction of these with the listed group properties exactly as required by existing literature. No step reduces a claim to a fitted parameter, self-referential definition, or unverified self-citation chain; the 2004 citation supplies background contrast rather than load-bearing justification for the new non-amenability statements. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the 2004 Runde-Spronk result and standard definitions of B(G), operator Connes-amenability, property (T), and the factorization property; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • standard math Standard definitions of the Fourier-Stieltjes algebra B(G), operator Connes-amenability, Kazhdan's property (T), almost periodic compactification, and the factorization property for groups.
    These are invoked to state the counterexamples and are drawn from prior literature including the 2004 result.

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