Weak Amenability and 2-weak Amenability of Beurling Algebras

Let Lω(G) be a Beurling algebra on a locally compact abelian group G. We look for general conditions on the weight which allows the vanishing of continuous derivations of Lω(G). This leads us to introducing vector-valued Beurling algebras and considering the translation of operators on them. This is then used to connect the augmentation ideal to the behavior of derivation space. We apply these results to give examples of various classes of Beurling algebras which are weakly amenable, 2-weakly amenable or fail to be even 2-weakly amenable. Let A be a Banach algebra, let n ≥ 0 be an integer, and let A(n) be the nth dual module of A when n > 0, and be A itself when n = 0. The algebra A is said to be weakly amenable if bounded derivations D : A → A∗ are inner, and it is said to be n-weakly amenable if bounded derivations D : A → A(n) are inner. The algebra A is permanently weakly amenable if it is n-weakly amenable for all n ≥ 1. The concept of weak amenability was first introduced by Bade, Curtis and Dales [2] for commutative Banach algebras, and was extended to the noncommutative case by B. E. Johnson [16] and it has been the object of many studies since (see for example, [11], [15], and [21] and references therein). Dales, Ghahramani and Grønbæk initiated the study of n-weakly amenable Banach algebras in [4], where they revealed many important properties of these algebras and presented some examples of them. For instance, they showed that C∗-algebras are permanently weakly amenable; the fact that was known for weakly amenable commutative Banach algebras [2, Theorem 1.5]. They also showed that group algebras are 2n+ 1-weakly amenable for all n > 0 (for more example see [11], [18], and [21]). Let Lω(G) be a Beurling algebra on a locally compact abelian group G. One can pose the question of whether Lω(G) is n-weakly amenable; in our case it means that each derivation from Lω(G) into L 1 ω(G) (n) is zero. The case of weak amenability has been studied in [2] and [13]. One major result states that l1 ω(Z) is weakly amenable if and only if infn ω(n)ω(−n) n = 0 [13]. From this, it can be easily deduced that l1 ω(G) is weakly amenable if 1991 Mathematics Subject Classification. Primary 43A20, 47B47.