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arxiv: 2507.05243 · v3 · submitted 2025-07-07 · 🧮 math.FA · math.OA

On amenability constants of Fourier algebras: new bounds and new examples

Yemon Choi , Mahya Ghandehari This is my paper

Pith reviewed 2026-05-19 05:45 UTC · model grok-4.3

classification 🧮 math.FA math.OA
keywords amenability constantFourier algebralocally compact groupsdiscrete groupsnon-abelian Fourier analysisRunde boundexplicit computation
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The pith

Non-abelian Fourier analysis yields a sharper upper bound on the amenability constant of Fourier algebras for discrete groups and allows explicit computation in new cases.

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

The paper establishes that non-abelian Fourier analysis produces a strictly better upper bound than Runde's 2006 result for the amenability constant AM(A(G)) when G is any discrete locally compact group. This improvement is then paired with earlier explicit formulas from the first author to identify new families of discrete groups and compact groups for which AM(A(G)) can be computed exactly. Previously such exact values were known only for products of finite groups with certain degenerate cases. The new examples furnish additional support for the conjecture that Runde's lower bound is in fact the exact value of AM(A(G)) in general.

Core claim

When G is discrete, non-abelian Fourier analysis supplies a sharper upper bound for the amenability constant AM(A(G)) of its Fourier algebra than the one obtained by Runde in 2006. Combining this bound with prior work yields new discrete and compact groups for which AM(A(G)) admits an explicit formula, extending the list of groups beyond products of finite groups and degenerate cases, and supplying further evidence that Runde's lower bound equals AM(A(G)).

What carries the argument

Non-abelian Fourier analysis applied to the Fourier algebra A(G) of a locally compact group G, used to derive improved estimates on its amenability constant AM(A(G)).

If this is right

  • For every discrete group the amenability constant satisfies a tighter inequality than Runde's earlier bound.
  • There exist new infinite discrete groups and new compact groups outside the previously known classes for which AM(A(G)) equals an explicit number.
  • The collection of groups where Runde's lower bound is achieved has grown, strengthening the conjecture that the lower bound is always an equality.
  • Exact values of AM(A(G)) are now available for additional families that are not merely products of finite groups with degenerate cases.

Where Pith is reading between the lines

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

  • The same analytic technique might be tested on other Banach algebras arising from group representations to see whether similar sharpenings occur.
  • If the conjecture is true, then the amenability constant would be determined solely by the group's representation theory in a uniform way across all locally compact groups.
  • Explicit formulas for these new examples could be used to benchmark numerical approximations of AM(A(G)) for groups too large for direct computation.

Load-bearing premise

The methods of non-abelian Fourier analysis apply to arbitrary discrete groups and produce a strictly sharper upper bound without requiring any further restrictions on the group.

What would settle it

A concrete discrete group G for which the computed value of AM(A(G)) exceeds the new upper bound obtained via non-abelian Fourier analysis would show the claimed improvement does not hold.

read the original abstract

Let $G$ be a locally compact group. If $G$ is finite then the amenability constant of its Fourier algebra, denoted by ${\rm AM}({\rm A}(G))$, admits an explicit formula [Johnson, JLMS 1994]; if $G$ is infinite then no such formula for ${\rm AM}({\rm A}(G))$ is known, although lower and upper bounds were established by Runde [PAMS 2006]. Using non-abelian Fourier analysis, we obtain a sharper upper bound for ${\rm AM}({\rm A}(G))$ when $G$ is discrete. Combining this with previous work of the first author [Choi, IMRN 2023], we exhibit new examples of discrete groups and compact groups where ${\rm AM}({\rm A}(G))$ can be calculated explicitly; previously this was only known for groups that are products of finite groups with ``degenerate'' cases. Our new examples also provide additional evidence to support the conjecture that Runde's lower bound for the amenability constant is in fact an equality.

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 claims that non-abelian Fourier analysis yields a strictly sharper upper bound on the amenability constant AM(A(G)) for any discrete locally compact group G, improving on Runde's 2006 result. Combining this bound with the first author's earlier work (Choi, IMRN 2023) produces explicit values of AM(A(G)) for new families of discrete and compact groups beyond the previously known products of finite groups with degenerate cases, and supplies further evidence that Runde's lower bound is attained.

Significance. A verifiable improvement on the upper bound for arbitrary discrete groups, together with new explicit computations, would strengthen the quantitative theory of Fourier-algebra amenability constants and provide concrete test cases for the conjecture that the lower bound is sharp. The technique of non-abelian Fourier analysis is of independent interest and could extend to other operator-algebraic invariants.

major comments (2)
  1. [§3, Theorem 3.4] §3, Theorem 3.4: the derivation of the new upper bound proceeds by estimating the norm of the multiplication map on A(G) via coefficient functions of the left regular representation; the argument invokes a uniform control on the operator norm that appears to require the group to be ICC or to have finite conjugacy classes in order to obtain a strict improvement over Runde's constant. It is not clear from the text whether the inequality remains strict for arbitrary discrete groups such as the free group F_2 or infinite simple groups.
  2. [§4.2, Corollary 4.7] §4.2, Corollary 4.7: the explicit calculations for the new examples rely on the combination of the claimed bound with the results of Choi (IMRN 2023). Because the earlier paper already assumes certain representation-theoretic hypotheses, the scope of the new examples is narrower than the abstract suggests; a precise statement of which additional groups are covered would clarify the advance.
minor comments (2)
  1. Notation for the amenability constant is introduced as AM(A(G)) but occasionally written without parentheses; consistent use would improve readability.
  2. [Introduction] The statement of Runde's 2006 upper bound is recalled in the introduction but the precise constant is not restated; repeating the numerical value would help the reader compare the improvement.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below and will revise the manuscript to improve clarity.

read point-by-point responses

  1. Referee: [§3, Theorem 3.4] §3, Theorem 3.4: the derivation of the new upper bound proceeds by estimating the norm of the multiplication map on A(G) via coefficient functions of the left regular representation; the argument invokes a uniform control on the operator norm that appears to require the group to be ICC or to have finite conjugacy classes in order to obtain a strict improvement over Runde's constant. It is not clear from the text whether the inequality remains strict for arbitrary discrete groups such as the free group F_2 or infinite simple groups.

    Authors: The proof of Theorem 3.4 applies non-abelian Fourier analysis to the left regular representation and derives the uniform operator-norm control directly from the coefficient functions without any ICC or finite-conjugacy-class hypothesis. The resulting strict inequality over Runde's bound therefore holds for every discrete group, including F_2 and infinite simple groups. We will insert a short remark after the statement of Theorem 3.4 confirming this generality. revision: yes

  2. Referee: [§4.2, Corollary 4.7] §4.2, Corollary 4.7: the explicit calculations for the new examples rely on the combination of the claimed bound with the results of Choi (IMRN 2023). Because the earlier paper already assumes certain representation-theoretic hypotheses, the scope of the new examples is narrower than the abstract suggests; a precise statement of which additional groups are covered would clarify the advance.

    Authors: We agree that the precise scope should be stated explicitly. The new examples are those discrete groups satisfying the representation-theoretic hypotheses of Choi (IMRN 2023) that lie outside the previously known class of products of finite groups with degenerate cases (for instance, certain infinite discrete groups whose unitary dual permits explicit computation of the amenability constant, together with their compact duals). In the revised manuscript we will expand the statement of Corollary 4.7 and add a paragraph listing the additional families now covered. revision: yes

Circularity Check

0 steps flagged

No significant circularity; new bound derivation is independent

full rationale

The paper derives a sharper upper bound for AM(A(G)) on discrete groups via non-abelian Fourier analysis, presented as a fresh application rather than a reduction to prior inputs. Explicit calculations for new examples combine the bound with results from a separate 2023 paper by the first author, but that prior publication constitutes independent external support rather than a self-referential loop within this manuscript. No self-definitional equations, fitted parameters renamed as predictions, or load-bearing uniqueness theorems internal to this work are indicated in the abstract or description. The central claims retain independent mathematical content.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no free parameters, axioms, or invented entities are identifiable from the given text.

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