Abstract
Let \({M\subseteq \mathbb{C}}\) be compact and \({K\subseteq M}\) closed, and let A(K, M) be the uniform algebra of all functions continuous on M and holomorphic in the interior K° of K. We present a constructive proof of Arens’ classical result that for \({(f_{1},\ldots,f_{n})\in A(K,M)^{n}}\) the Bézout equation \({\sum_{j=1}^{n} a_{j}f_{j}=1}\) has a solution in A(K, M) if and only if the functions f j have no common zero in K. We shall also consider matrix-valued Bézout equations.
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Dedicated to Professor Michael von Renteln on the occasion of his 70th birthday
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Mortini, R., Rupp, R. A solution to the Bézout equation in A(K) without Gelfand theory. Arch. Math. 99, 49–59 (2012). https://doi.org/10.1007/s00013-012-0395-x
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DOI: https://doi.org/10.1007/s00013-012-0395-x