$(\sigma,\tau)$-amenability of $C^*$-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 $\big(\tau({\mathcal A}),\sigma({\mathcal A})\big)$-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 $\big(\tau({\mathcal A}),\sigma({\mathcal A})\big)$-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 $C^*$-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