Vol. 92, No. 1, 1981

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On the closed ideals in A(W)

Charles Madison Stanton

Vol. 92 (1981), No. 1, 199–209

This paper is about the ideal theory of the algebra of functions continuous on the closure and holomorphic in the interior of a domain on a compact Riemann surface. The description of the closed ideals in the disc algebra is shown to apply to an ideal whose hull meets the boundary of the domain in a finite union of analytic arcs. The canonical factorization into inner and outer functions in the disc is replaced by a potential theoretic decomposition theorem, thus allowing essentially the same description to be carried over. The basically local nature of the problem is used to reduce it to the previously known ideal theory of a compact bordered Riemann surface. This reduction is facilitated by a factorization theorem that is proved by potential theoretic methods.

Mathematical Subject Classification 2000
Primary: 30H05
Secondary: 30F99, 46J15
Received: 28 August 1978
Revised: 8 April 1980
Published: 1 January 1981
Charles Madison Stanton