$L$-analytic mappings in the disk algebra

Abstract

It is shown that two classes of function transformations coincide when the transformations take place within the disk algebra. The first class is that of the L-analytic mappings. These are the ones given locally by power series: fg(f-fo)n The second class is that of locally pointwise mappings. A mapping f-[f] is pointwise if it has the form (I [fI (x)=I *(x, f (x)). It is a by-product of the disk algebra investigation that if a set X has certain topological properties, then every locally pointwise mapping in C(X) is continuous. In 1943 E. R. Lorch introduced an analytic theory for mappings whose domain and range lie in a commutative Banach algebra with identity [3]. Let A be such an algebra, and let D be an open connected subset of A. A mapping (D: D-?A is L-analytic, that is analytic in the sense of Lorch if in a neighborhood of each g e D we have a power series expansion D[f I= E. gn(f-g)n. The series is to converge in the norm of A, and the coefficients gnl are elements of A (which depend on g). To study L-analytic mappings it is standard procedure to use a technique similar to the Gelfand transform. Let M be a complex homomorphism of A onto the complex numbers C. We say (D: D--A quotients on D with respect to M if there is an ordinary holomorphic function (M which is defined on M(D) and satisfies (DM o M=M o (D. Call (DM the quotient function of (D at M. If (D quotients on D with respect to every M, we say (D quotients on D. On certain domains, for instance on balls, an Lanalytic mapping will always quotient. For more on quotient functions see [1]. The fact that a mapping quotient has an important interpretation when A is a algebra of functions on a space X. By we mean that A determines the topology of X and that every homomorphism of A onto C is given by evaluation at a point of X. The terminology is due to Rickart [4]. Let A be a natural algebra of functions on a space X, and let E,i be the evaluation functional E$(f)=f (x). To say (D: D--A quotients on D is to say that for each x E X we have a quotient function @D that satisfies Received by the editors August 30, 1972. AMS (MOS) subject classifications (1970). Primary 30A96, 30A98; Secondary 46J15. ? American Mathematical Society 1973