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  "body": "\n## Definition\n\nLet $A$ be a complex Banach algebra.\nA *Banach $A$-bimodule* is a Banach space $X$ with bounded bilinear left and right actions $A \\times X \\to X$ and $X \\times A \\to X$ that make $X$ an $A$-bimodule.\nThe [[dual space|4A82S5FVRT]] $X^*$ is then a Banach $A$-bimodule with $(a \\cdot f)(x) = f(x \\cdot a)$ and $(f \\cdot a)(x) = f(a \\cdot x)$ for $a \\in A$, $f \\in X^*$, $x \\in X$.\nA bounded linear map $D \\colon A \\to X^*$ is a *[[derivation|3J89REV9WQ]]* if $D(ab) = a \\cdot D(b) + D(a) \\cdot b$ for all $a, b \\in A$.\nIt is *inner* if there is $f \\in X^*$ with $D(a) = a \\cdot f - f \\cdot a$ for all $a \\in A$.\nThe algebra $A$ is *amenable* if, for every Banach $A$-bimodule $X$, every bounded derivation $A \\to X^*$ is inner.\n\n## Remarks\n\nIn cohomological terms, $A$ is amenable if and only if $H^1(A, X^*) = 0$ for every Banach $A$-bimodule $X$.\n\nExample: $A = \\C$.\nLet $D \\colon \\C \\to X^*$ be a derivation and put $u = D(1)$.\nFrom $D(1) = D(1 \\cdot 1) = 1 \\cdot u + u \\cdot 1$ one gets $1 \\cdot u \\cdot 1 = 0$.\nPut $f = 1 \\cdot u - u \\cdot 1$.\nThen $1 \\cdot f - f \\cdot 1 = 1 \\cdot u - 2 (1 \\cdot u \\cdot 1) + u \\cdot 1 = u = D(1)$, and by linearity $D(\\lambda) = \\lambda \\cdot f - f \\cdot \\lambda$ for all $\\lambda \\in \\C$.\nSo every bounded derivation is inner.\n\nNon-example: let $\\mathbb{F}_2$ be the free group on two generators, which is not an [[amenable group|CEAT9BESKK]].\nBy Johnson's theorem, $L^1(G)$ is amenable if and only if the locally compact group $G$ is amenable.\nSo $\\ell^1(\\mathbb{F}_2)$ is not amenable: some bounded derivation into a dual bimodule is not inner.\n\nA C*-algebra is amenable if and only if it is [[nuclear|D2M4VXJ7PA]].\n",
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