The structure of ideals and point derivations in Banach algebras of Lipschitz functions

Abstract

Introduction. The main object of study in this paper is the space of all bounded complex-valued functions defined on a metric space (X, d) which satisfy a Lipschitz condition with respect to the metric d. With a suitable norm this collection of functions becomes a Banach algebra. This Banach algebra, which we shall call a Lipschitz algebra, will be denoted by Lip(X, d). It is the structure of Lip(X, d) with which we are primarily concerned here. Although the notion of Lipschitz continuity is very old and Lipschitz functions have been studied for many years, interest in the Banach space and Banach algebra theory of Lipschitz functions has not developed until quite recently. Very little is known about the Banach space properties of Lip (X, d). The following papers constitute all the work known to this writer which has been done in this area. Mirkil [9] and de Leeuw [4] have considered the space of periodic Lipschitz functions on the real line (or in other words, Lipschitz functions defined on the circle group). Mirkil was interested in the relation between the Lipschitz condition and the translation properties of functions, the fact that the functions were defined on a group being vital. de Leeuw was more directly concerned with the Banach space structure, his paper involving the study of certain dual spaces, extreme points, and isometries. In the long paper of Glaeser [6], a short chapter, somewhat off the main theme of his work, is devoted to the space of n-times continuously differentiable functions defined on a subset of Emn whose nth derivatives satisfy a Lipschitz condition. It is shown as a side result in the paper of Arens and Eells [1] that Lip(X,d) is always the dual space of some normed linear space. The only work known to us which treats Lip(X,d) as a Banach algebra was done by S. B. Myers [11]. His paper [11] is a summary of results, the proofs never published because of his untimely death in 1955. His interest in Lipschitz algebras was connected with the process of differentiation, and this interest we pursue in Chapter III of the present study. We supply proofs of many of the unproved statements of Myers, and in some cases extend his results to more general settings.