PICARDS THEOREM, MITTAG-LEEFLER METHODS, AND CONTINUITY OF CHARACTERSON FRECHET ALGEBRAS

The general question of continuity of characters on commutative Freche t algebras can be reduced to the question of continuity of characters on some ''test algebras''. We discuss the algebraic structure of some quotients U/I where U is one of these test algebras. The notion of Pic ard-Borel algebra, related to the classical Picard theorem, plays an i mportant role in these investigations. We also use the Mittag-Leffler theorem to exhibit large semigroups in;quotients of the form A/I where A is a commutative unital Frechet algebra and I a dense, countable un ion of closed prime ideals of ii. We point out new algebraic obstructi ons to the construction of discontinuous characters on U related to th e Picard theorem, and relate to extension properties of joint spectra of finite families of a quotient of U a question about iteration of Bi eberbach mappings raised in 1986 by P. G. Dixon and the author.