Ideals and Boundaries in Algebras of Holomorphic Functions
We investigate the spectrum of certain Banach algebras. Propertieslike 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 coincidewith 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 inU 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.
Source Type:Doctoral Dissertation
Keywords:MATHEMATICS; maximal ideal space; the Gleason problem; generalized Shilov boundaries; Nebenhülle; the Koszul complex; Banach algebras of holomorphic functions
Date of Publication:01/01/2006