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  "body": "\n## Summary\n\nAn amenable Banach algebra is a Banach algebra for which every bounded [[derivation|3J89REV9WQ]] into every dual bimodule is inner.\nBarry Johnson introduced the notion.\nHe proved that for a locally compact group $G$, the group algebra $L^1(G)$ is amenable exactly when $G$ is an [[amenable group|CEAT9BESKK]].\nFor C*-algebras, amenability is equivalent to being [[nuclear|D2M4VXJ7PA]].\nIt is one of the main finiteness conditions in the cohomology of Banach algebras.\n\n## Explanations\n\n### L1 human\n\nA derivation is a linear map that obeys the product rule $D(ab) = a D(b) + D(a) b$.\nSome derivations come from a simple formula, $D(a) = a f - f a$ for a fixed $f$; these are called inner.\nAn algebra is amenable if, in a whole class of settings, every derivation is of this simple kind.\nFor example, the complex numbers $\\C$ form an amenable Banach algebra, and so does every algebra $C(X)$ of continuous functions on a compact Hausdorff space.\nThe name comes from amenable groups, whose group algebras are exactly the amenable ones.\n",
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