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 $\geq 3/2$, extending result [9] who proved this holds for finite non-abelian groups. Our lower bound, which known to be best possible, improves on results by previous authors and answers question raised [16]. To do this, we study minorant constant, related anti-diagonal in $G\times G$, was implicitly used Runde’s work but hitherto not studied depth. main novelty an explicit formula when countable virtually abelian group, terms Plancherel measure $G$. As further applications, characterize those groups where attains minimal value present some examples support conjecture always coincides with constant.