<p>We investigate the spectrum of certain Banach algebras. Properties</p><p>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 ⊂⊂ C<sup>n</sup> 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.</p><p>For a domain D ⊂⊂ C<sup>n</sup> where the boundary of the Nebenhülle coincide</p><p>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.</p><p>If the boundary of an open set U is smooth we show that there exist points in</p><p>U such that the maximal ideals over those points are generated by the coordinate functions.</p><p>An example is given of a Riemann domain, Ω, spread over C<sup>n</sup> where the fibers over a point p ∈ Ω consist of m > n elements but the maximal ideal over p is generated by n functions.</p>
| Identifer | oai:union.ndltd.org:UPSALLA/oai:DiVA.org:umu-675 |
| Date | January 2006 |
| Creators | Carlsson, Linus |
| Publisher | Umeå University, Mathematics and Mathematical Statistics, Umeå : Matematik och matematisk statistik |
| Source Sets | DiVA Archive at Upsalla University |
| Language | English |
| Detected Language | English |
| Type | Doctoral thesis, monograph, text |
| Relation | Doctoral thesis / Umeå University, Department of Mathematics, 1102-8300 ; 33 |
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