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A constructive proof of the Nullstellensatz for subalgebras of \(A(K)\). (English) Zbl 0735.30040

Séminaire de mathématique de Luxembourg, Trav. Math. 3, 45-49 (1991).
[For the entire collection see Zbl 0728.00002.]
Let \(A(K)\) be the Banach algebra of all continuous complex valued functions on a compact set \(K\subset\mathbb{C}\), which are analytic in the interior \(K^ 0\) of \(K\). The authors prove, in an elementary and constructive way, the following version of the Nullstellensatz for subalgebras \(A\) of \(A(K)\) (it can also be considered as a form of the Corona theorem): Assume that \(A\) satisfies the conditions: (1) Each continuous function on \(\partial K\) can be approximated uniformly by rational functions with poles off \(\partial K\), (2) A contains the polynomials, (3) \((f-f(z_ 0))/(z-z_ 0)\in A\) whenever \(f\in A\) and \(z_ 0\in K^ 0\). Then, if \(f_ 1,\dots,f_ N\in A\) have no common zero in \(K\) then there exist functions \(g_ 1,\dots,g_ N\in A\) such that \(\sum_{i=1}^ N g_ i f_ i\) has no zero in \(K\).
Reviewer: T.Krasiński

MSC:

30H05 Spaces of bounded analytic functions of one complex variable
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces

Citations:

Zbl 0728.00002
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