Central Beurling algebras: Weak amenability of the central Beurling algebras on [FC]$^-$ groups

Abstract

We study weak amenability of central Beurling algebras $ZL^1(G,ω)$. The investigation is a natural extension of the known work on the commutative Beurling algebra $L^1(G,ω)$. For [FC]$^-$ groups we establish a necessary condition and for [FD]$^-$ groups we give sufficient conditions for the weak amenability of $Z\L1o$. For a compactly generated [FC]$^-$ group with the polynomial weight $ω_α(x) = (1 + |x|)^α$, we prove that $ZL^1(G,ω_α)$ is weakly amenable if and only if $α< 1/2$.

To appear in the Michigan Math J