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

This paper establishes a sharper upper bound for the amenability constant of the Fourier algebra of discrete groups and uses it to calculate explicit values for new classes of discrete and compact groups, thereby providing further evidence for the conjecture that Runde's lower bound is an equality.

The Gist

Here is an explanation of the paper "On amenability constants of Fourier algebras: new bounds and new examples," translated into simple language with creative analogies.

The Big Picture: Measuring the "Chaos" of a Group

Imagine you have a group of people (a mathematical Group). They can interact, swap places, and follow rules. Mathematicians study these groups by turning them into a "soundtrack" of functions called a Fourier Algebra. Think of this algebra as a complex musical score that captures not just the melody (the shape of the group) but also the harmony and rhythm (how the group members interact).

Some groups are very orderly (like a choir singing in perfect unison), while others are chaotic (like a mosh pit). In math, we call the orderly ones Amenable.

The authors of this paper are trying to answer a very specific question: How "messy" is a group?

To do this, they use a ruler called the Amenability Constant (let's call it the "Messiness Score").

• If the score is low (close to 1), the group is very orderly.
• If the score is high (or infinite), the group is chaotic.
• If the score is infinite, the group is too chaotic to be "amenable" at all.

The Problem: We Have a Ruler, But It's Broken

For a long time, mathematicians knew how to calculate this Messiness Score for small, finite groups (like a group of 5 people). They had a perfect formula.

However, for infinite groups (like the integers, which go on forever), the formula broke. We knew the score existed, but we didn't know exactly what it was. We only had a rough estimate:

• Lower Bound: "It's at least this high."
• Upper Bound: "It's no higher than this."

The gap between the lower and upper bounds was huge. It was like trying to guess the price of a house knowing only that it's "more than $100" and "less than $10 million."

The Breakthrough: Sharper Rulers and New Examples

The authors, Choi and Ghandehari, did two main things to fix this:

1. They Sharpened the Upper Bound

They invented a new, much tighter ruler for a specific type of group (discrete, virtually abelian groups).

• The Analogy: Imagine you are trying to guess the height of a giant. Before, you said, "He's between 5 feet and 100 feet tall." That's not very helpful.
• The New Result: The authors said, "Actually, based on his shoe size and the way he walks, he is definitely between 6 feet and 6 feet 2 inches."
• Why it matters: This new, sharper upper bound allowed them to calculate the exact Messiness Score for many groups where it was previously impossible.

2. They Found New "Test Subjects"

They applied this new ruler to two specific, interesting groups:

• The Integer Heisenberg Group: A group based on whole numbers.
• The p-adic Heisenberg Group: A group based on a strange type of number system used in advanced number theory.

For these groups, they didn't just guess the score; they calculated the exact number.

• The Result: They found that for these groups, the Messiness Score is exactly equal to a specific value derived from the group's structure ($p - 1 + 1/p$).

The Big Conjecture: The "Perfect Match" Theory

The most exciting part of the paper is what these results suggest about a famous guess (Conjecture 1.2).

Mathematicians have a theory that says: The Messiness Score is always exactly equal to a specific "Anti-Diagonal" measurement.

• Think of the "Anti-Diagonal" as a shadow cast by the group.
• The theory says: The Shadow Length = The Messiness Score.

For decades, this was just a theory. We knew it worked for small groups, but we didn't know if it held up for the complex, infinite ones.

• The Paper's Contribution: By calculating the exact Messiness Score for their new examples and showing it matches the "Shadow Length" perfectly, the authors provided strong new evidence that the theory is true.

Why Should You Care?

You might wonder, "Who cares about the Messiness Score of a group of numbers?"

1. It connects different worlds: This work bridges the gap between pure algebra (groups) and analysis (functions and calculus). It shows that the structure of a group dictates the behavior of its functions in a very precise way.
2. It solves a 30-year-old puzzle: The paper answers questions that have been open since the 1990s.
3. It builds a foundation: By proving these specific cases, they give mathematicians the tools to solve the problem for all groups in the future.

Summary in a Nutshell

• The Goal: Measure how orderly or chaotic a mathematical group is.
• The Obstacle: For infinite groups, we only had rough estimates, not exact numbers.
• The Solution: The authors created a sharper measuring tool and used it to find the exact "Messiness Score" for two new, complex groups.
• The Discovery: In these new cases, the Messiness Score perfectly matches a theoretical prediction, suggesting a deep, universal rule about how groups behave.

It's like finding two new planets and discovering they orbit exactly where the laws of physics predicted they should, giving us confidence that our understanding of the universe is correct.

Technical Summary

Here is a detailed technical summary of the paper "On amenability constants of Fourier algebras: new bounds and new examples" by Y. Choi and M. Ghandehari.

1. Problem Statement and Context

The paper addresses the calculation of the amenability constant, denoted $AM(A(G))$, for the Fourier algebra $A(G)$ of a locally compact group $G$.

• Background: A Banach algebra is amenable if and only if its amenability constant is finite. While the amenability of $A(G)$ is fully characterized (it is amenable iff $G$ is virtually abelian), the exact value of $AM(A(G))$ is generally unknown for infinite groups.
• Known Results:
  • For finite groups, Johnson (1994) provided an explicit formula: $AM(A(G)) = \frac{1}{|G|} \sum_{\pi \in \widehat{G}} (d_\pi)^3$.
  • For general groups, Runde (2006) established bounds: $1 \leq AD(G) \leq AM(A(G)) \leq \maxdeg(G)$, where$AD(G)$is the Fourier-Stieltjes norm of the anti-diagonal and$\maxdeg(G)$ is the supremum of degrees of irreducible unitary representations.
• The Gap: No general formula exists for infinite groups. Furthermore, it is unknown whether $AM(A(G_1 \times G_2)) = AM(A(G_1))AM(A(G_2))$ (Question 1.1).
• The Conjecture: The authors focus on Conjecture 1.2: $AM(A(G)) = AD(G)$ for every locally compact group $G$. If true, this would imply the multiplicativity of the amenability constant for products.

2. Methodology

The authors employ a combination of non-abelian Fourier analysis, operator space theory, and structural group theory.

• Non-Abelian Fourier Analysis: The core technical tool is the use of the Plancherel theorem for second-countable unimodular Type I groups. They express norms in $A(G)$ and tensor products $A(G) \hat{\otimes} A(G)$ in terms of operator-valued Fourier coefficients and the Plancherel measure $\nu$.
• Reduction to Countable Subgroups: A key methodological step is proving that for any discrete group $G$, there exists a countable subgroup $\Gamma$ such that $AM(A(G)) = AM(A(\Gamma))$. This allows the authors to apply Plancherel theory (which typically requires second-countability) to general discrete groups.
• Sharp Upper Bounds: The authors refine Runde's upper bound. Instead of simply using $\maxdeg(G)$, they derive a sharper bound involving the commutator subgroup $[G, G]$ and the set of 1-dimensional representations.
• Inverse Limits: For compact groups (specifically profinite groups), they utilize the structure of inverse limits of finite groups to approximate the amenability constant.

3. Key Contributions and Results

A. Theoretical Improvements

1. Sharper Upper Bound for Discrete Groups (Theorem 4.1):
   For a discrete, virtually abelian group $G$, the authors prove:
   $$AM(A(G)) \leq 1 + (\maxdeg(G) - 1) \left( 1 - \frac{1}{|[G, G]|} \right)$$
   This improves upon Runde's bound $\maxdeg(G)$ and recovers Johnson's finite group formula when $G$ is finite. The proof relies on estimating the norm of the inverse of the map $J: A(G) \hat{\otimes} A(G) \to A(G \times G)$ restricted to the diagonal.

2. Countable Saturation (Theorem 1.3):
   For any discrete group $G$, there exists a countable subgroup $\Gamma \leq G$ such that $AM(A(G)) = AM(A(\Gamma))$. This reduces the study of amenability constants for arbitrary discrete groups to the countable case.

3. Multiplicativity and Functoriality:
   The paper confirms that if Conjecture 1.2 holds for countable groups, it holds for all discrete groups (Corollary 1.4). It also establishes that the class of groups satisfying the conjecture is closed under finite products (Corollary 1.5).

B. New Explicit Examples

The authors construct new families of groups where $AM(A(G))$ can be calculated explicitly, confirming Conjecture 1.2 in cases not covered by previous "degenerate" examples (products of finite groups or abelian groups).

1. Groups with Two Degrees of Irreps (Theorem 1.8):
   If $G$ is a countable group where the set of degrees of irreducible representations is $\{1, d\}$, then:
   $$AM(A(G)) = AD(G) = 1 + (d - 1) \left( 1 - \frac{1}{|[G, G]|} \right)$$
   This generalizes the finite group formula to infinite discrete groups.

2. Minimal Value Characterization (Theorem 1.9):
   For a discrete non-abelian group $G$, $AM(A(G)) = 3/2$ if and only if $|G : Z(G)| = 4$. This answers a specific question from previous literature regarding the minimal possible value for non-abelian groups.

3. Reduced Heisenberg Groups (Theorem 1.7):
   The authors define $H_{rp}(S)$ as a quotient of the Heisenberg group over a ring $S$ (specifically $S = \mathbb{Z}$ and $S = \mathbb{Z}_p$).

   • Discrete Case: $H_{rp}(\mathbb{Z})$ is a discrete group.
   • Compact Case: $H_{rp}(\mathbb{Z}_p)$ is a compact (profinite) group.
   • Result: For both cases, $AM(A(G)) = AD(G) = p - 1 + p^{-1}$.
   • Significance: These groups are not isomorphic to products of finite groups and AD-maximal groups, proving that the class of groups satisfying the conjecture is broader than previously thought.

4. Significance

• Evidence for Conjecture 1.2: The paper provides strong empirical evidence supporting the conjecture that $AM(A(G)) = AD(G)$. By calculating exact values for non-trivial infinite groups (both discrete and compact) and showing they match the lower bound $AD(G)$, the authors narrow the gap between known bounds.
• Resolution of Open Questions: The results provide a positive answer to Question 1.1 for specific new classes of groups and resolve Question 6.2 from a previous paper (Choi, 2023) for discrete groups with index 4 centers.
• Methodological Advancement: The derivation of the sharp upper bound (Theorem 4.1) without relying on Johnson's finite group formula (which is unavailable for infinite groups) is a significant technical achievement. It demonstrates that the "sharper" bound is a structural property of virtually abelian groups, not just a finite phenomenon.
• Future Directions: The paper highlights that the Plancherel measure plays a crucial role in the proof but disappears from the final formula, suggesting a deeper structural reason for the equality $AM(A(G)) = AD(G)$. It opens the door to investigating profinite groups and extending the sharp bound to non-discrete virtually abelian groups.

In summary, this paper advances the quantitative theory of amenability for Fourier algebras by establishing new sharp bounds, proving reduction theorems for discrete groups, and providing the first explicit calculations for non-trivial infinite groups, thereby strongly supporting the conjecture that the amenability constant equals the anti-diagonal norm.