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We prove that three pairwise disjoint, convex sets can be found, all congruent to a set of the form , such that their union has a non-trivial polynomial convex hull. This shows that not all holomorphic functions on the interior of the union can be approximated by polynomials in the open-closed topology.
For a domain let be the holomorphic functions on and for any let . Denote by the set of functions with the property that there exists a sequence of functions such that is a nonincreasing sequence and such that . By denote the set of functions with the property that there exists a sequence of functions such that is a nondecreasing sequence and such that . Let and let and be bounded -domains of holomorphy in and respectively. Let , and . We prove that the...
We study the spectrum of certain Banach algebras of holomorphic functions defined on a domain Ω where ∂̅-problems with certain estimates can be solved. We show that the projection of the spectrum onto ℂⁿ equals Ω̅ and that the fibers over Ω are trivial. This is used to solve a corona problem in the special case where all but one generator are continuous up to the boundary.
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