Function algebras

Abstract

Introduction. By a function algebra I shall mean a collection A of complex-valued functions on a set X such that the (pointwise) sum and the product of two functions in A are again in A. We shall always suppose that A contains the constant functions so that A becomes an algebra with unit over the field of complex numbers. A function algebra A is called self-ad joint if the complex conjugate of each function in A is again in A, and the theory of self-ad joint algebras is quite different from that of non-self-adjoint algebras. An example of a selfadjoint function algebra is given by the algebra of all continuous complex-valued functions on a topological space X. A thorough description of the theory of such algebras is given in the book by Gilman and Jerison [27].