An explicit minorant for the amenability constant of the Fourier algebra

Abstract

We show that if a locally compact group $G$ is non-abelian then the amenability constant of its Fourier algebra is $\ge 3/2$, 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.

v5: AMS-LaTeX, 30 pages. To appear in IMRN. Only differences from v4 are some extra corrections of typos and minor tweaks for consistency of style. This does not incorporate the changes made by the journal's copy-editors

Cited by in corpus (2)

• Small values and forbidden values for the Fourier antidiagonal constant of a finite group
• On amenability constants of Fourier algebras: new bounds and new examples