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arXiv · 2012.14413

An explicit minorant for the amenability constant of the Fourier algebra

Y. Choi
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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.

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Y. Choi. 2020-12-28. An explicit minorant for the amenability constant of the Fourier algebra. https://doi.org/10.1093/imrn%2Frnac348

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