An abstract potential theory with continuous kernel

Abstract

l Introduction.. In the study of complex function algebras, it is a standard technique to consider the functions as being defined on the spectrum (maximal ideal space) of the algebra. In other words, one routinely replaces a function algebra A by its Gelfand representation A. Recall that the Gelfand representation of a Banach algebra is just the standard representation of any normed space A as a family of functionals on A*. Each xeA is represented as the functional x on points FeA* defined by x(F) = F{x). The Gelfand representation simply restricts the domain of x to the very small set consisting of those FeA* which are multiplicative (i.e., to the homomorphisms of the algebra A). Of course this restriction is necessary if A is to be again an algebra. However, a fair amount of structure accrues to the representation by virtue of this restriction (cf.