An explicit minorant for the amenability constant of the Fourier algebra

TL;DR

This paper establishes a lower bound of 3/2 for the amenability constant of Fourier algebras of non-abelian groups, providing explicit formulas for certain groups and exploring when this bound is attained.

Contribution

It introduces an explicit minorant for the amenability constant, extending previous results and answering a longstanding question about its minimal value.

Findings

Lower bound of 3/2 for non-abelian groups' Fourier algebra amenability constant

Explicit formula for the minorant in countable virtually abelian groups

Characterization of groups where the minorant attains its minimal value

Abstract

We show that if a locally compact group $G$ is non-abelian then the amenability constant of its Fourier algebra is $\geq 3/2$, extending a result of Johnson (JLMS, 1994) who proved that this holds for finite non-abelian groups. Our lower bound, which is known to be best possible, improves on results by previous authors and answers a question raised by Runde (PAMS, 2006). To do this we study a minorant for the amenability constant, related to the anti-diagonal in $G\times G$, which was implicitly used in Runde's work but hitherto not studied in depth. Our main novelty is an explicit formula for this minorant when $G$ is a countable virtually abelian group, in terms of the Plancherel measure for $G$. As further applications, we characterize those non-abelian groups where the minorant attains its minimal value, and present some examples to support the conjecture that the minorant always coincides with the amenability constant.