Johnson's theorem on amenable group algebras

Remarks

The idea of the proof: suppose $G$ is amenable and let $D \colon L^1(G) \to X^*$ be a bounded derivation. The action of $L^1(G)$ on $X^*$ extends to an action of the point masses $\delta_x$, $x \in G$, and $D$ extends to them as well. Averaging the bounded family $x \mapsto \delta_{x^{-1}} \cdot D(\delta_x)$ of elements of $X^*$ with the invariant mean gives an element of $X^*$ that implements $D$ as an inner derivation. Conversely, if $L^1(G)$ is amenable, one applies the definition to a suitable bimodule built from $L^\infty(G)$ and constants; the element that makes the derivation inner yields an invariant mean.

Example: abelian groups and compact groups are amenable, so $\ell^1(\Z)$, $L^1(\R)$ and $L^1(G)$ for compact $G$ are amenable Banach algebras.

Non-example: the free group $F_2$ on two generators is not amenable, so $\ell^1(F_2)$ is not an amenable Banach algebra.

The theorem was proved by B. E. Johnson in 1972, in the work where he introduced amenable Banach algebras; their name comes from this theorem.

Going deeper

For a student

The theorem lets one translate questions about amenable groups into Banach algebra cohomology, and it motivated the whole study of amenable Banach algebras. For the measure algebra the situation is different: $M(G)$ is amenable only when $G$ is discrete and amenable (Dales, Ghahramani and Helemskii). On the Fourier side, amenability of $G$ corresponds instead to a bounded approximate identity in the Fourier algebra, by Leptin's theorem. For C*-algebras, amenability is equivalent to nuclearity, and the reduced group C*-algebra of a discrete group is nuclear exactly when the group is amenable.