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In this paper, for a discrete group $G=mathbb{Z} astmathbb{Z}_n $ and a weight function of polynomial $ omega_ alpha $, we show that the Burling algebra $ ell^1(G, omega_ alpha) $ is not weakly amenable and dihedral group $D_ infty=mathbb{Z}_2astmathbb{Z}_2 $ is amenable. We also show that for a continuous weight function $ omega $ under certain conditions on group $ G $, if the Burling algebra $ ell^1(G, omega ) $ is weakly amenable then $ omega $ is bounded.

In this paper we introduce the notion of φ-commutativity for a Banach algebra A, where φ is a continuous homomorphism on A and study the concept of φ-weak amenability for φ-commutative Banach algebras. We give an example to show that the class of φ-weakly amenable Banach algebras is larger than that of weakly amenable commutative Banach algebras. We characterize φ-weak amenability of φ-commutative Banach algebras and prove some hereditary properties. Moreover we verify some of the previous available results about commutative weakly amenable Banach algebras, for φ-commutative φ-weakly amenable Banach algebras.

For a Banach algebra A, A00 is (􀀀 1)-weakly amenable if A0 is a Banach A00-bimodule and H1(A00; A0) = f0g. In this paper we prove some important properties of this notion, for instance if A00 is (􀀀 1)-weakly amenable then A is essential and there is no non-zero point derivation on A. We also give some examples, namely, the second dual of every C-algebras is (􀀀 1)-weakly amenable. Finally, we study the relationships between the (􀀀 1)-weakly amenability of A00 and the weak amenability of A00 or A.

For two Banach algebras $A$ and $B$ and a Banach $(A,B)$-module $X$, let $\mathrm{Tri}(A,X,B)$ be the corresponding triangular Banach algebra. In this paper, among other things, we give some characterizations of $n$-weak amenability of $\mathrm{Tri}(A,X,B)$, without any assumptions on $A$ and $B$. We then apply our characterizations to obtain some of the known results in the literature with fewer assumptions and provide a direct proof. To illustrate our results we present some examples, where old results in the literature are not effective in identifying their $n$-weak amenability.