Vol. 59, No. 2, 1975

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Bounded analytic functions on a class of open Riemann surfaces

Charles Madison Stanton

Vol. 59 (1975), No. 2, 557–565

In this paper some function theoretic properties of an open Riemann surface are related to a condition on the exhaustion of the surface by finite Riemann surfaces. The class of Myrberg surfaces is introduced; these are certain branched covering surfaces of the unit disc. The exhaustion condition is used to distinguish those Myrberg surfaces on which the bounded analytic functions separate points. A complete description is given of the ways in which the space of bounded analytic functions on a Myrberg surface can degenerate. The exhaustion condition is stated in terms of the Green’s function; it is already known to be equivalent to a function theoretic condition on the fundamental group of the surface. This latter condition is shown to imply that the surface is an open subset of the spectrum of its Banach algebra of bounded analytic functions.

Mathematical Subject Classification
Primary: 30A98
Received: 1 February 1974
Revised: 2 May 1975
Published: 1 August 1975
Charles Madison Stanton