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An explicit minorant for the amenability constant of the Fourier algebra

Research output: Contribution to Journal/MagazineJournal articlepeer-review

<mark>Journal publication date</mark>30/11/2023
<mark>Journal</mark>International Mathematics Research Notices
Issue number22
Pages (from-to)19390–19430
Publication StatusPublished
Early online date22/06/23
<mark>Original language</mark>English


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 [9] 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 [16]. 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.