On absolutely convergent Dirichlet series

Abstract

satisfying 1 b,j < oo, if and only if f(s) is bounded away from zero in the half-plane o?0. This discovery amounts to a determination of the spectrum of the Banach-algebra element associated with f(s), and it thus makes available for the theory of ordinary Dirichlet series the well-known theorem of Gelfand on analytic functions of Banach-algebra elements (see ?24D of [7]) and its generalization, the Silov-Arens-Calderon theorem (Theorem 3.3 of [i]). This paper is intended to show that the theorem of Hewitt and Williamson, together with a similar theorem for general Dirichlet series En=0O a,e-xn8, can be deduced by an elementary argument from a theorem due to Phillips (stated below, in a weakened form, as Theorem 1). The possibility of such an argument arises from the circumstance that the Banach algebra considered by Hewitt and Williamson (and likewise each of the algebras 2{r introduced below) can be enlarged in a certain way without decreasing the spectra of the original elements. No attempt however will be made to examine the maximal ideal spaces of the algebras which occur (the reader who wishes to pursue the subject is referred to the writings [4; 5] of Hewitt and Zuckerman): the object here is to show what can be done without introducing new machinery.