Ideals and boundaries in Algebras of Holomorphic functions

• Main Author: Carlsson, Linus
• Format: Doctoral Thesis
• Language: English
• Published: Umeå universitet, Institutionen för matematik och matematisk statistik 2006
• Subjects: maximal ideal space; the Gleason problem; generalized Shilov boundaries; Nebenhülle; the Koszul complex; Banach algebras of holomorphic functions; MATHEMATICS; MATEMATIK
• Online Access: http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-675; http://nbn-resolving.de/urn:isbn:91-7264-011-1

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, Ω, spread over Cn where the fibers over a point p ∈ Ω consist of m > n elements but the maximal ideal over p is generated by n functions.