Algebras of Finite Differential Order and the Operational Calculus

Abstract

The notions of topological spaces and of continuous functions are well established in mathematics. The related concepts of differentiable spaces and of differentiable functions have not as yet been introduced in a comparable broad and definitive form, as their study is classically confined mostly to locally euclidean spaces. The present work arose from wishes to extend part of the known results of the theory of differentiable manifolds, due to Whitney, to a larger class of spaces which are not required to be locally euclidean a priori. As a motivation to the question studied here, let us mention that an n-differentiable structure on a topological space X is definable by a topological algebra A of real continuous functions on X, A being the algebra of n-differentiable real functions on X and the topology of A being that of uniform convergence of a function and all its derivatives of order ? n on compact sets. A has to satisfy suitable requirements, among them the following two conditions. If d e X, let M(e) be the closed ideal of all f e A such that f (a) = 0. Then all the powers Ml(e), i > n + 1, are identical and A/Mn+,(;) is a local algebra whose maximal ideal has (n+-1)power equal to 0. Moreover, A is locally convex (? 3) with respect to the collection {Mn+,(f)}, a e X. Such a local convexity assumption expresses the local nature of A (rather its infinitesimal nature). Another property of A is that it has an operational calculus with ordinary differentiable functions, i.e., that f e A implies rp(f) e A for every n-differentiable real function q on the real line R, the mapping ((p, f ) --'p(f ) being continuous; and similarly for functions q of several variables. Since this second property of A can be shown to follow from the first, provided A is complete, we consider the question, when do suitable local convexity assumptions imply existence of an operational calculus? the answer to which is given by the theorem stated in ? 7. This article is devoted to the proof of the assertion that, if C is a category of pure separated algebras whose radicals are nilpotent and n > 0 is an integer, in order that every topological algebra which is