Amenable Banach algebra

Builds on Derivation Continuous dual space

Definition

Let $A$ be a complex Banach algebra. A 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. The dual space $X^*$ 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 $a \in A$ , $f \in X^*$ , $x \in X$ . A bounded linear map $D \colon A \to X^*$ is a derivation if $D(ab) = a \cdot D(b) + D(a) \cdot b$ for all $a, b \in A$ . It is inner if there is $f \in X^*$ with $D(a) = a \cdot f - f \cdot a$ for all $a \in A$ . The algebra $A$ is amenable if, for every Banach $A$ -bimodule $X$ , every bounded derivation $A \to X^*$ is inner.

Remarks

In cohomological terms, $A$ is amenable if and only if $H^1(A, X^*) = 0$ for every Banach $A$ -bimodule $X$ .

Example: $A = \C$ . Let $D \colon \C \to X^*$ be a derivation and put $u = D(1)$ . From $D(1) = D(1 \cdot 1) = 1 \cdot u + u \cdot 1$ one gets $1 \cdot u \cdot 1 = 0$ . Put $f = 1 \cdot u - u \cdot 1$ . Then $1 \cdot f - f \cdot 1 = 1 \cdot u - 2 (1 \cdot u \cdot 1) + u \cdot 1 = u = D(1)$ , and by linearity $D(\lambda) = \lambda \cdot f - f \cdot \lambda$ for all $\lambda \in \C$ . So every bounded derivation is inner.

Non-example: let $\mathbb{F}_2$ be the free group on two generators, which is not an amenable group. By Johnson's theorem, $L^1(G)$ is amenable if and only if the locally compact group $G$ is amenable. So $\ell^1(\mathbb{F}_2)$ is not amenable: some bounded derivation into a dual bimodule is not inner.

A C*-algebra is amenable if and only if it is nuclear.

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Tag	`ET71R6CSAZ`
URI	`m2://ET71R6CSAZ`
Kind	definition
Writing	W0 stub
MSC	46H20, 43A07
Defines	Amenable Banach algebra
Uses	Derivation, Continuous dual space
Changed	2 November 2026

Amenable Banach algebra

Definition

Remarks

Reading path

Further reading

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