The Gleason problem has been proven to be a complicated issue to tackle. In this thesiswe will conclude that a domain, Ω ⊂ R𝑛, has Gleason 𝑅-property at any point 𝑝 ∈ Ω, where 𝑅(Ω) ⊂ 𝐶∞(Ω) is the ring of functions that are real analytic in 𝑝. First, we investigate function spaces and give them fitting norms. Afterwards, we build a bump function that is then used to construct a smooth partition of unity on R𝑛. Finally, we show that some of the functionspaces, introduced earlier, have the Gleason property. Ultimately, we use our smooth partition of unity in order to prove that the statement above holds for domains in R2. Subsequently, with the same reasoning one can prove that the statement also holds for domains Ω ⊂ R𝑛.

<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 ⊂⊂ 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.</p><p>For a domain D ⊂⊂ Cⁿ 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 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.</p>

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 &gt; n elements but the maximal ideal over p is generated by n functions.