Approximate semi-amenability of Banach algebras

Abstract

Let $\mathfrak{A}$ be a Banach algebra, and $\mathcal{X}$ a Banach $\mathfrak{A}$-bimodule. A bounded linear mapping $\mathcal{D}:\mathfrak{A}\to \mathcal{X}$ is approximately semi-inner derivation if there eixist nets $({\xi }_{\alpha }{)}_{\alpha }$ and $({\mu }_{\alpha }{)}_{\alpha }$ in $\mathcal{X}$ such that, for each $a\in \mathfrak{A}$, $\mathcal{D}(a)={\lim }_{\alpha }(a.{\xi }_{\alpha }-{\mu }_{\alpha }.a)$. $\mathfrak{A}$ is called approximately semi-amenable if for every Banach $\mathfrak{A}$-bimodule $\mathcal{X}$, every $\mathcal{D}\in {\mathcal{Z}}^1(\mathfrak{A},{\mathcal{X}}^{\ast })$ is approximtely semi-inner. There are some Banach algebras which are approximately semi-amenable, but not approximately amenable. In this manuscript, we investigate some properties of approximate semi-amenability of Banach algebras. Also in Theorem \ref{ee} we prove the approximate semi-amenability of Segal algebras on a locally compact group $G$.

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• Approximate semi-amenability of Banach algebras