A characterization of \(C(K)\) among function algebras on a Riemann surface. (English) Zbl 1192.46048

Let \(K\) be a compact subset of an open Riemann surface. Denote by \(C(K)\) and \(A(K)\) the algebra of all complex-valued continuous functions on \(K\) and its subalgebra consisting of those functions which are holomorphic on the interior of \(K\), respectively.
The author presents necessary and sufficient conditions for a function algebra containing \(A(K)\) to be all of \(C(K)\). Using these results, several conditions are given on a complex-valued function \(f\) so that the uniformly closed subalgebra generated by \(A(K)\) and \(f\) is all of \(C(K)\). In particular, the author considers the cases when \(f\) is continuously differentiable or harmonic on the interior of \(K\), or it is harmonic with respect to \(A(K)\). Finally, sufficient conditions are given for the algebra \(A(K)\) to be a maximal subalgebra of \(C(K)\). The presented results generalize several of A. J. Izzo’s former results from compact subsets of the complex plane to compact subsets of a Riemann surface.


46J10 Banach algebras of continuous functions, function algebras
32A38 Algebras of holomorphic functions of several complex variables
30F15 Harmonic functions on Riemann surfaces
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
Full Text: Euclid


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