Ideals and boundaries in Algebras of Holomorphic functions

Sammanfattning: We investigate the spectrum of certain Banach algebras. Properties like generators of maximal ideals and generalized Shilov boundaries are studied. In particular we show that if the ?-equation has solutions in the algebra of bounded functions or continuous functions up to the boundary of a domain D ?? Cn then every maximal ideal over D is generated by the coordinate functions. This implies that the fibres over D in the spectrum are trivial and that the projection on Cn of the n ? 1 order generalized Shilov boundary is contained in the boundary of D.For a domain D ?? Cn where the boundary of the Nebenhülle coincide with the smooth strictly pseudoconvex boundary points of D we show that there always exist points p ? D such that D has the Gleason property at p.If the boundary of an open set U is smooth we show that there exist points in U such that the maximal ideals over those points are generated by the coordinate functions.An example is given of a Riemann domain, ?, spread over Cn where the fibers over a point p ? ? consist of m > n elements but the maximal ideal over p is generated by n functions.