---
tag: 6P9AE21Q7X
kind: concept
name: Amenable Banach algebra
msc: [46H20, 43A07]
sources:
  - url: https://en.wikipedia.org/wiki/Amenable_Banach_algebra
same_as:
  wikidata: Q4742386
  wikipedia: Amenable Banach algebra
---

## Summary

An amenable Banach algebra is a Banach algebra for which every bounded [[derivation|3J89REV9WQ]] into every dual bimodule is inner.
Barry Johnson introduced the notion.
He proved that for a locally compact group $G$, the group algebra $L^1(G)$ is amenable exactly when $G$ is an [[amenable group|CEAT9BESKK]].
For C*-algebras, amenability is equivalent to being [[nuclear|D2M4VXJ7PA]].
It is one of the main finiteness conditions in the cohomology of Banach algebras.

## Explanations

### L1 human

A derivation is a linear map that obeys the product rule $D(ab) = a D(b) + D(a) b$.
Some derivations come from a simple formula, $D(a) = a f - f a$ for a fixed $f$; these are called inner.
An algebra is amenable if, in a whole class of settings, every derivation is of this simple kind.
For 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.
The name comes from amenable groups, whose group algebras are exactly the amenable ones.