A theory of analytic functions in Banach algebras

Abstract

Introduction. The present paper is concerned with the general problem of extending the classical theory of analytic functions of a complex variable. This question received the attention of Hilbert and F. Riesz, and probably goes back to Volterra. More recently N. Dunford, L. Fantappie, I. Gelfand, E. R. Lorch, A. D. Michal, and A. E. Taylor have contributed to the subject (see bibliography). Our approach differs from most of the others in two main respects, namely, in the type of domain and range of the functions and in the definition of analyticity. We consider functions which have for their domains and ranges subsets of an abstract commutative with unit and we use a definition of analyticity introduced by E. R. Lorch [1]. It is known [4] that a function analytic by this definition is differentiable in the Frechet sense but not every Frechet-differentiable function on a commutative is analytic in the Lorch sense. Accordingly, the Lorch theory is the richer. For the most part, the development of the primary aspects of the Lorch theory parallels that of the classical theory. Interesting departures occur in the more advanced stages. As one would expect, the Cauchy integral theorem and formula occupy a central position and yield the Taylor expansion in the usual way. With Lorch's work as a foundation, we have extended the theory to include a study of Laurent expansions and analytic continuation. There are also some results on the zeros of polynomials over the algebra, on rational functions and their integrals, and on the singularities of analytic functions. Although the objective of this investigation was essentially analytical, we have also obtained results of an algebraic-topological character (e.g. distribution of singular elements). This was a natural outcome of the algebraic character of the techniques used: 1. Basic concepts. A set, B, of elements (denoted by Latin letters a, b, c, x, y, z, ) is a Banach algebra if (1) B is an over the complex numbers (denQted by greek letters), (2) B is a complex space, and (3) the norm satisfies the inequality Ilabil <||a||||b||. In addition, we assume that multiplication is commutative and that B contains a unit element, e, with I|e|| = 1i. We recall without proof some of the salient facts about algebras. Multiplication is continuous in both factors together in the metric topology