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$(σ,τ)$-amenability of $C^*$-algebras

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