$(\sigma ,\tau )$-amenability of $C^{\ast }$-algebras

Abstract

Suppose that $\mathcal{A}$ is an algebra, $\sigma ,\tau :\mathcal{A}\to \mathcal{A}$ are two linear mappings such that both $\sigma (\mathcal{A})$ and $\tau (\mathcal{A})$ are subalgebras of $\mathcal{A}$ and $\mathcal{X}$ is a $(\tau (\mathcal{A}),\sigma (\mathcal{A}))$-bimodule. A linear mapping $D:\mathcal{A}\to \mathcal{X}$ is called a $(\sigma ,\tau )$-derivation if $D(ab)=D(a)\cdot \sigma (b)+\tau (a)\cdot D(b)(a,b\in \mathcal{A})$. A $(\sigma ,\tau )$-derivation $D$ is called a $(\sigma ,\tau )$-inner derivation if there exists an $x\in \mathcal{X}$ such that $D$ is of the form either $D_x^{-}(a)=x\cdot \sigma (a)-\tau (a)\cdot x(a\in \mathcal{A})$ or $D_x^{+}(a)=x\cdot \sigma (a)+\tau (a)\cdot x(a\in \mathcal{A})$. A Banach algebra $\mathcal{A}$ is called $(\sigma ,\tau )$-amenable if every $(\sigma ,\tau )$-derivation from $\mathcal{A}$ into a dual Banach $(\tau (\mathcal{A}),\sigma (\mathcal{A}))$-bimodule is $(\sigma ,\tau )$-inner. Studying some general algebraic aspects of $(\sigma ,\tau )$-derivations, we investigate the relation between amenability and $(\sigma ,\tau )$-amenability of Banach algebras in the case when $\sigma ,\tau$ are homomorphisms. We prove that if $\mathfrak{A}$ is a $C^{\ast }$-algebra and $\sigma ,\tau$ are *-homomorphisms with $\ker (\sigma )=\ker (\tau )$, then $\mathfrak{A}$ is $(\sigma ,\tau )$-amenable if and only if $\sigma (\mathfrak{A})$ is amenable

9 pages; to appear in Georgian Math. J