Vol. 99, No. 1, 1982

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The classification of uniform algebras on plane domains

William Robin Zame

Vol. 99 (1982), No. 1, 231–247

Let Ω be an open subset of the complex plane. Denote by 𝒪(Ω) the algebra of all holomorphic functions on Ω, equipped with the topology of uniform convergence on compact set. The object of this paper is to provide a complete classification of all the closed subalgebras of 𝒪(Ω) which contain the polynomials, and apply this classification to several concrete problems, including localness of these algebras, continuity of homomorphisms, and number of generators. It should be emphasized that no assumptions are made as to the connectivity of Ω. In fact, in the cases of most interest, Ω will not be connected and some of the connected components of Ω will not be simply connected.

Mathematical Subject Classification 2000
Primary: 46J15
Secondary: 30H05
Received: 9 January 1980
Revised: 7 October 1980
Published: 1 March 1982
William Robin Zame