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 ZL 1 (G, ω).For a compactly generated [FC] -group with 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.