article

Title: Weak amenability of Fourier algebras on compact groups

Authors: Brian E. Forrest, Ebrahim Samei and Nico Spronk

Issue: Volume 58 (2009), Issue 3, 1379-1394

Abstract:

$We give for a compact group $G$, a full characterization of when its Fourier algebra $\falg$ is weakly amenable: when the connected component of the identity $G_e$ is abelian. This condition is also equivalent to the hyper-Tauberian property for $\falg$, and to having the anti-diagonal $\check{\Del} = \{(s,s^{-1}):s\in G\}$ be a set of spectral synthesis for $\falgg$. We extend our results to some classes of non-compact, locally compact groups, including small invariant neighbourhood groups and maximally weakly almost periodic groups. We close by illustrating a curious relationship between amenability and weak amenability of $\falg$ for compact $G$, and (operator) amenability and (operator) weak amenability of $\fdelg$, an algebra defined by the authors in [B.E. Forrest and P.J. Wood, \emph{Cohomology and the operator space structure of the Fourier algebra and its second dual}, Indiana Math. J. \bftext{50} (2001), 1217--1240].$