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

Choi, Yemon and Ghandehari, Mahya (2026) On amenability constants of Fourier algebras : new bounds and new examples. Journal of the London Mathematical Society. ISSN 0024-6107 (In Press)

Text (amag-newbound_rev1)
amag-newbound_rev1.pdf - Accepted Version
Available under License Creative Commons Attribution.
Download (494kB)

Abstract

Let $G$ be a locally compact group. If $G$ is finite then the amenability constant of its Fourier algebra, denoted by ${\rm AM}({\rm A}(G))$, admits an explicit formula [Johnson, JLMS 1994]; if $G$ is infinite then no such formula for ${\rm AM}({\rm A}(G))$ is known, although lower and upper bounds were established by Runde [PAMS 2006]. Using non-abelian Fourier analysis, we obtain a sharper upper bound for ${\rm AM}({\rm A}(G))$ when $G$ is discrete. Combining this with previous work of the first author [Choi, IMRN 2023], we exhibit new examples of discrete groups and compact groups where ${\rm AM}({\rm A}(G))$ can be calculated explicitly; previously this was only known for groups that are products of finite groups with "degenerate"' cases. Our new examples also provide additional evidence to support the conjecture that Runde's lower bound for the amenability constant is in fact an equality.