Continuity of Derivations in F-Algebras

Abstract

In [1] Johnson considered commutative semisimple F-algebras with identity which are also Q-algebras. (A Q-algebra is an algebra in which the set of invertible elements is open.) For an algebra A of this type, Johnson proved that any derivation of A into A^, the algebra of all Gelfand transforms of elements of A, is continuous. The difficulty with this theorem is that the standard examples of F-algebras with derivations are not Q-algebras. In particular, neither the algebra CG (0, 1), nor the algebra Hol (Q), each with the standard F-algebra topology, is a Q-algebra. Here Co (0, 1) denotes the algebra of all complex valued infinitely differentiable functions on (0, 1), and Hol (Q) is the algebra of all functions which are holomorphic on an open subset Q of the complex plane. In fact, one often turns to F-algebras in order to avoid the restrictions (such as every element having a compact spectrum) placed on an algebra by requiring the algebra to be a Q-algebra. Tihe purpose of this note is to show how Johnson's techniques can be used to prove that every derivation of an F-algebra A into the algebra A is continuous, where it is only assumed that A is commutative, semisimple, and has an identity. From this fact, we can conclude that every derivation of A into A' is continuous, since any such derivation can be factored through A. In particular, we have that any derivation of a Frechet function algebra into itself is continuous. A will denote a commutative semisimple F-algebra with identity. For an element a of A, aA will denote the Gelfand transform of the element a, and A' will denote the algebra of all Gelfand transforms of elements of A. If B is an F-algebra, then we will use Spec B to denote the space of all continuous homomorphisms of B onto the complex numbers, where the space of homomorphisms is given the Gdelfand topology. We fix a sequence {Bn} of Banach algebras such that A is the inverse limit of the Banach algebras