Document Type : Research Paper

Author

Department of Mathematics‎, ‎Faculty of Basic Sciences‎, ‎Imam Ali University‎, ‎Tehran‎, ‎Iran

Abstract

‎Let ${\A}$ be a Banach space and ${\lambda}$ be a non-zero fixed element of ${\A}^{\ast}$(dual space of ${\A}$) with non-zero kernel‎. ‎Defining algebra product in $\A$ as $a\cdot b=\lambda(a)b$ for $a,b\in {\A}$‎, ‎we show that ${\A}$ is a $(2m-1)$-weakly amenable Banach algebra but not $2m$-weakly amenable for any $m\in{\N}$‎. ‎Furthermore‎, ‎we show the converse of the statement [2,~Proposition\,1.4.(ii)] for a non-unital Banach algebra $\A$‎, ‎if $\A$ is weakly amenable then $\A^{\#}$ is weakly amenable‎" ‎does not hold‎.

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