Amenability and measure algebras

We show that the measure algebra M(G) of a locally compact group G is amenable as a Banach algebra if and only if G is discrete and amenable as a group. Our contribution is to resolve a conjecture by proving that M(G) is not amenable unless G is discrete. Indeed, we prove a much stronger result: the measure algebra of a nondiscrete, locally compact group has a non-zero, continuous point derivation at a certain character of the algebra.