Function Algebras

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]. My own interest in function algebras arose from the study of algebras of analytic functions on some sort of an analytic space X. These algebras are of course very far from being self-ad joint. A considerable amount of effort has gone into the study of certain of these algebras and of the relationship between algebraic properties of such an algebra and the analytic structure of the underlying space X. This includes work by Bers [8], Bishop, [lO; 12], Chevalley and Kakutani [43], Edwards [22], Heins [29], Helmer [30], Henriksen [33], Rudin [57], Wermer [70; 71 ] and myself [54; 55]. One of the principal purposes of the present discussion is to give a general treatment of function algebras into which many of the above results may be fitted. Another purpose is to illustrate the application of methods from the theory of functions of several complex variables to derive results about non-self-ad joint function algebras. Excellent examples of such applications are given in the papers by Arens and Calderon [2 ] and Rossi [52], and in §7 I use a theorem of Arens and mine to illustrate these techniques. This theorem describes the one-dimensional cohomology of the spectrum of a Banach algebra in terms of the group of invertible elements of the algebra. It generalizes to arbitrary commutative Banach algebras the theorems of Bruschlinsky [18] and Eilenberg [23], describing the one-dimensional cohomology of a compact set X in terms of C(X).