Subalgebras of finite codimension in the algebra of analytic functions on a Riemann surface

Bruce Stephen Lund

Vol. 51 (1974), No. 2, 495–497

DOI: 10.2140/pjm.1974.51.495

Abstract

Let R be a finite open Riemann surface with boundary Γ. We set R = R ∪ Γ and let A(R) denote the algebra of functions which are continuous on R and analytic on R. Suppose A is a uniform algebra contained in A(R). The main result of this paper shows that if A contains a function F which is analytic in a neighborhood of R and which maps R in a n-to-one manner (counting multiplicity) onto {z : |z|≦ 1}, then A has finite codimension in A(R).

Mathematical Subject Classification 2000

Primary: 30A98

Secondary: 46J15

Milestones

Received: 22 February 1973

Revised: 25 June 1973

Published: 1 April 1974