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.

Keywords