Vol. 42, No. 3, 1972

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Algebras of analytic functions in the plane

William Robin Zame

Vol. 42 (1972), No. 3, 811–819

Let X be a compact subset of the complex plane and let A be an algebra of functions analytic near X which contains the polynomials and is complete in its natural topology. This paper is concerned with determining the spectrum of A and describing A in terms of its spectrum. It is shown that the spectrum of A is formed from the disjoint union of certain compact subsets of C (suitably topologized) by making certain identifications. A is closed under differentiation exactly when no identifications need be performed, and then A admits a simple, complete description. In particular, if X is connected, then the completion of A is merely the restriction to X of the algebra of all functions analytic near the union of X with some of the bounded components of C X.

Mathematical Subject Classification 2000
Primary: 46J15
Secondary: 30A98
Received: 21 May 1971
Published: 1 September 1972
William Robin Zame