Document Type : Research Paper

Authors

• T. Yazdanpanah
• I. Mozzami Zadeh

Department of Mathematics, Persian Gulf University, Bushehr, 75168, Iran

Abstract

Let $\mathcal{A}$ be a Banach algebra, $\sigma$ be continuous homomorphism on $\mathcal{A}$ with $\overline{\sigma (\mathcal{A})}=\mathcal{A}$. The bounded linear map $D:\mathcal{A}\to {\mathcal{A}}^{\ast }$ is $\sigma$-derivation, if
$$D(ab)=D(a)\sigma (b)+\sigma (a)D(b)(a,b\in \mathcal{A}).$$
We say that A is $\sigma$-weakly amenable, when for each bounded derivation $D:\mathcal{A}\to {\mathcal{A}}^{\ast }$, there exists $a^{\ast }\in {\mathcal{A}}^{\ast }$ such that $D(a)=\sigma (a)a^{\ast }-a^{\ast }\sigma (a)$. For a commutative Banach algebra $\mathcal{A}$, we show $\mathcal{A}$ is $\sigma$-weakly amenable if and only if every $\sigma$-derivation from $\mathcal{A}$ into a $\sigma$-symmetric Banach $\mathcal{A}$-bimodule $X$ is zero. Also, we show that a commutative Banach algebra $\mathcal{A}$ is $\sigma$-weakly amenable if and only if $A^{\mathrm{\#}}$ is ${\sigma }^{\mathrm{\#}}$-weakly amenable, where ${\sigma }^{\mathrm{\#}}(a+\alpha )=\sigma (a)+\alpha$.

Keywords

• Banach algebra
• $\sigma$-derivation
• $\sigma$-weak amenability