Characterising Derivations from the Disc Algebra to Its Dual

Although derivations from Banach algebras to particular coefficient modules have been much studied, interest has usually focused on the existence or otherwise of non-trivial bounded derivations, rather than their characterisation. Even in the special case of derivations from a commutative Banach algebra to its dual (as in [2] or [5] for example), there are comparatively few examples where the space of such derivations is explicitly characterised or parametrised. For uniform algebras, very little is known: the only examples with a complete characterisation are the trivial uniform algebras C(X) – and these have no nonzero bounded derivations into any dual C(X)-bimodule. In this paper we provide the first example of a non-trivial uniform algebra – namely, the disc algebra – where the space of derivations to its dual can be completely characterised. For the disc algebra, this is equivalent to characterising those complex Borel measures μ on the unit circle, for which the bilinear functional (f, g) 7→ ∫