TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 203, 1975 ANALYTIC STRUCTURE IN SOMEANALYTIC FUNCTION ALGEBRAS BY WILLIAM R. ZAME(') ABSTRACT. A complete description is given of the analytic structure of maximal dimension in the spectra of a wide class of concrete function algebras generated by analytic functions. A connection is also given with point deriva- tions on such algebras. I. Introduction. Let B be a function algebra acting on its spectrum (maximal ideal space) Sp B, with Shilov boundary TB. According to a well- known result of Rossi [15], the functions in B obey a weak maximum modulus principle on Sp B - TB. One of the principal themes in the theory of function algebras is the "explanation" of this analytic-type behavior by the existence of analytic structure in Sp B. More precisely, let us say that an analytic variety (of dimension n) in Sp B is a pair (X, r) where X is an analytic variety (of pure dimension n), r: X —► Sp B is a continuous map, and b ° r is analytic for each b e B. Then the question is: what are the nontrivial analytic varieties in SpS? Although Stolzenberg's example [16] shows that one does not have the existence of nontrivial analytic structure in all situations, a number of global and local conditions which imply the existence of analytic structure have been found by Gleason [11], Browder [8], Clayton [9] and others. A different approach, due to Bishop [3], which depends upon the existence of functions in B with finite fibers, leads to the construction of 1-dimensional analytic structure in Sp B. Bishop's ideas have been expanded and exploited by Stolzenberg [17], Bjork [5], [6], Alexander [1], [2], the author [20], and others, and have proven very use- ful in dealing with questions of approximation. In this paper, we shall be concerned with the classification of high-dimen- sional analytic structure. For a fairly wide class of concrete function algebras, those which contain a sufficiently rich collection of analytic functions, we are able to completely describe the analytic structure of maximal dimension which Received by the editors August 27, 1973 and, in revised form, February 6, 1974. AMS (MOS) subject classifications (1970). Primary 46J10; Secondary 32E2S, 32E30. Key words and phrases. Analytic function algebras, analytic structure, differentiably stable algebras, point derivations. (!) Supported in part by NSF Grant P037961 and by a grant from the Research Foun- dation, State University of New York. Copyright © 1975, American Mathematical Society 215 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use