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Ideals and boundaries in Algebras of Holomorphic functions
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
2006 (English)Doctoral thesis, monograph (Other academic)
Abstract [en]

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.

Place, publisher, year, edition, pages
Umeå: Umeå universitet , 2006. , 97 p.
Doctoral thesis / Umeå University, Department of Mathematics, ISSN 1102-8300 ; 33
Keyword [en]
maximal ideal space, the Gleason problem, generalized Shilov boundaries, Nebenhülle, the Koszul complex, Banach algebras of holomorphic functions
National Category
URN: urn:nbn:se:umu:diva-675ISBN: 91-7264-011-1OAI: diva2:144207
Public defence
2006-02-17, MIT-huset, MA121, Umeå universitet, UMEÅ, 10:15
Available from: 2006-01-24 Created: 2006-01-24 Last updated: 2012-10-11Bibliographically approved

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Carlsson, Linus
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