Let A be a semisimple Banach algebra with a linear automorphism σ and let δ: I → A be a σ-derivation, where I is an ideal of A. Then Φ(δ)(I ∩ σ(I)) = 0, where Φ(δ) is the separating space of δ. As a consequence, if I is an essential ideal then the σ-derivation δ is closable. In a prime C*-algebra, we show that every σ-derivation defined on a nonzero ideal is continuous. Finally, any linear map on a prime semisimple Banach algebra with nontrivial idempotents is continuous if it satisfies the σ-derivation...

In this paper we extend the notion of $n$-weak amenability of a Banach algebra $𝒜$ when $n\in \mathbb{N}$. Technical calculations show that when $𝒜$ is Arens regular or an ideal in ${𝒜}^{**}$, then ${𝒜}^{*}$ is an ${𝒜}^{\left(2n\right)}$-module and this idea leads to a number of interesting results on Banach algebras. We then extend the concept of $n$-weak amenability to $n\in \mathbb{Z}$.