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On uniform algebras containing A(K). (English. Russian original) Zbl 0591.46045
Russ. Math. Surv. 40, No. 2, 205-206 (1985); translation from Usp. Mat. Nauk 40, No. 2, 169-170 (1985).
Let K be a compact subset of the complex plane, \(K^ 0\) be the interior of K which consists of a finite number of components of connectedness \(G_ 1,...,G_ n\) and \(\partial K\) be the boundary of K. Let A(K) be the uniform algebra of all continuous functions on K which are holomorphic in \(K^ 0.\)
The description of such uniform algebras on \(\partial K\) which contain A(K) as subalgebra is given. It is stated that A(K\(\setminus \cup^{n}_{i=1}G_ i)\) coincides with the uniform algebra on \(\partial K\) generated by the functions from A(K) and by the functions \((z-z_ 1)^{-1},...,(z-z_ n)^{-1}\), where \(z_ i\in G_ i\) for each i.
Reviewer: M.Abel

MSC:
46J10 Banach algebras of continuous functions, function algebras
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
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