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.
| Identifer | oai:union.ndltd.org:UPSALLA1/oai:DiVA.org:umu-675 |
| Date | January 2006 |
| Creators | Carlsson, Linus |
| Publisher | Umeå universitet, Institutionen för matematik och matematisk statistik, Umeå : Umeå universitet |
| Source Sets | DiVA Archive at Upsalla University |
| Language | English |
| Detected Language | English |
| Type | Doctoral thesis, monograph, info:eu-repo/semantics/doctoralThesis, text |
| Format | application/pdf |
| Rights | info:eu-repo/semantics/openAccess |
| Relation | Doctoral thesis / Umeå University, Department of Mathematics, 1102-8300 ; 33 |
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