Spaces of functions with values in a Banach algebra

Abstract

Introduction. In the following pages certain spaces of abstract-valued functions are examined. Throughout the paper A will denote a commutative Banach algebra. In analogy to the group algebra L'(G) of a locally compact Abelian group G, that is, the space of absolutely integrable complex-valued functions on G, we form the set B1 =B'(G, A) of Bochner integrable functions defined to A from G. Bl is first of all a Banach space and it becomes a commutative Banach algebra if multiplication of two elementsf, gcBl is defined by the convolution formula, (f*g) (x) =f Gf(xy)g(y-1)dy. For the theory of the Bochner integral we shall rely mainly oni the presentatioIn in Hille's book [2, pp. 35-49]. Although the development there uses the Lebesgue measure for finite dimensional Euclidean spaces, the theorems which we shall need hold as well for more general measure spaces, in particular for a locally compact group with Haar measure. The calculus for the generalized convolution carries over directly from that for numerical functions and will be assumed. We note that the convolution of a function in B' and a funiction in L'(G) is well defined and in Bl. To the greatest possible extent the notation and definitions are those of Loomis [3]. Maximial means regular maximal ideal throughout. Special conventions are as follows: f =-f(x), g = g(x), * * denote elemenlts of B1. For the most part complex-valued functions are assigned Greek letters regardless of the set on which they are defined. X is always used for the characteristic function of a set and A is the Haar measure on G. 5TB denotes the maximal ideal space of B1, ORA that of A, and G, the character group of G, is the maximal ideal space of Ll. Typical elements are MB, M, and a respectively. Subscripts distinguish the various norms. For example, |f|I IB =_ff(X)llAdx, fEB'. For any complex or A-valued function on G and xCG, the subscript x applied to the function denotes its translate by x. The symbol used with