Spectrum of certain Banach algebras and $\overline{\partial}$-problems
Volume 90 / 2007
Annales Polonici Mathematici 90 (2007), 51-58
MSC: 32A65, 32W05, 46J20.
DOI: 10.4064/ap90-1-4
Abstract
We study the spectrum of certain Banach algebras of holomorphic functions defined on a domain ${\mit\Omega}$ where $\overline\partial$-problems with certain estimates can be solved. We show that the projection of the spectrum onto $\mathbb C^n$ equals $\,\overline{\!{\mit\Omega}}$ and that the fibers over ${\mit\Omega}$ 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.
