An amenable, radical Banach algebra

We give an example of an amenable, radical Banach algebra, relying on results from non-abelian harmonic analysis due to H. Leptin, D. Poguntke and J. Boidol. Let A be a Banach algebra, and let E be a Banach A-module. A bounded linear map D : A → E is called a a derivation if D(ab) = a.Db+ (Da).b (a, b ∈ A). A derivation D : A → E is said to be inner if Da = x.a− a.x (a ∈ A) for some x ∈ E. For any Banach A-module E, its dual space E is naturally equipped with a Banach A-module structure via 〈x, a.φ〉 := 〈x.a, φ〉 and 〈x, φ.a〉 := 〈a.x, φ〉 (a ∈ A, x ∈ E, φ ∈ E). We can now give the definition of an amenable Banach algebra: Definition 1 A Banach algebra A is amenable if, for each Banach A-module E, every derivation D : A → E is inner. The notion of amenability for Banach algebras was introduced by B. E. Johnson in [Joh]. A locally compact group G is called amenable if it possesses a translation-invariant mean, i.e. if there is a linear functional φ : L(G) → C satisfying φ(1) = ‖φ‖ = 1 and φ(δx ∗ f) = φ(f) (x ∈ G, f ∈ L (G)). For instance, all abelian and all compact groups are amenable. For further information, see the monograph [Pat]. In [Joh], B. E. Johnson proved the following fundamental theorem which provides the motivation for Definition 1: Theorem 2 Let G be a locally compact group. Then G is amenable if and only if L1(G) is amenable.