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 the polynomial weight ωα(x) = (1 + |x|) α , we prove that ZL 1 (G, ωα) is weakly amenable if and only if α < 1/2.

Center of Banach algebra valued Beurling algebras

“…Note that [18] generalizes the results of [1,2] and in these three articles the major focus was on studying the amenability and weak amenability properties. The idea behind the proofs given in [18,2] is to use a projection from L 1 (G) onto Z(L 1 (G)).…”

“…Note that [18] generalizes the results of [1,2] and in these three articles the major focus was on studying the amenability and weak amenability properties. The idea behind the proofs given in [18,2] is to use a projection from L 1 (G) onto Z(L 1 (G)). Ingenious construction of one such projection is given in [18] which is somewhat different from the usual averaging technique used while working with [FIA] − groups.…”

“…For two algebras A and B, Z(A) ⊗ Z(B) = Z(A ⊗ B), where Z(C) denotes the center of algebra C. If A and B are Banach algebras, then it is natural ask whether Z(A ⊗ γ B) is isometrically isomorphic to Z(A) ⊗ γ Z(B), ⊗ γ being the Banach space projective tensor product. It is known to be true if A and B are C * -algebras (see [8,Theorem 5.1]) and if A = B = L 1 (G) for any [FC] − group G (see [18,Lemma 2.1]).…”

“…The idea behind the proofs given in [18,2] is to use a projection from L 1 (G) onto Z(L 1 (G)). Ingenious construction of one such projection is given in [18] which is somewhat different from the usual averaging technique used while working with [FIA] − groups. We used this technique in [7,Theorem 4.13] and provided an affirmative answer to this question if A is a unital Banach algebra and B = L 1 (G), for specific classes of groups.…”

Non-weakly amenable Beurling algebras

“…Note that since {x : g 1,y (x) = 0} ⊂ U yH (y ∈ Y ) and the family {U yH} y∈Y is locally finite, the sum in the definition of g 1 has only finitely many non-zero terms in a neighborhood of every point. This implies that g 1 is well-defined, and because each g 1,y is continuous, g 1 is also continuous on G. From (27) and the local finiteness of {U yH} y∈Y it follows that (24) holds. The inclusion (25) also holds since it holds for each g 1,y .…”