Thus by a ‘derivation’ D we shall mean here an operator satisfying the first identity (the ‘derivation identity’) for all x,y E ‘8, and (except in the next two introductory paragraphs) lying in %3(a); and likewise for the others. E denotes the identity element in b(a). The implications of the above equations have been widely studied, though generally not for bounded operators on a Banach algebra. The importance of the first identity needs no emphasis; in differential geometry, for example, derivations on algebras of differentiable functions are of basic importance. A derivation D on an algebra with identity e has no inverse, since D(e) = 0; however, writing I = D-l formally makes I an antiderivation. The formula for integration by parts can be put in the form of the antiderivation identity, which may thus be thought of as a generalization of it: cf. F. V. Atkinson [I], where antiderivations arise by the specialization 0 = 0 in Baxter’s identity