Amenability properties of the central Fourier algebra of a compact group

Abstract

We let the central Fourier algebra, $\operatorname{ZA}(G)$, be the subalgebra of functions $u$ in the Fourier algebra $\mathrm{A}(G)$ of a compact group, for which $u(xyx^{-1})=u(y)$ for all $x$, $y$ in $G$. We show that this algebra admits bounded point derivations whenever $G$ contains a non-Abelian closed connected subgroup. Conversely when $G$ is virtually Abelian, then $\operatorname{ZA}(G)$ is amenable. Furthermore, for virtually Abelian $G$, we establish which closed ideals admit bounded approximate identities. We also show that $\operatorname{ZA}(G)$ is weakly amenable, in fact hyper-Tauberian, exactly when $G$ admits no non-Abelian connected subgroup. We also study the amenability constant of $\operatorname{ZA}(G)$ for finite $G$ and exhibit totally disconnected groups $G$ for which $\operatorname{ZA}(G)$ is non-amenable. In passing, we establish some properties related to spectral synthesis of subsets of the spectrum of $\operatorname{ZA}(G)$.