A Proof of the Corona Conjecture for Finite Open Riemann Surfaces

By a finite open Riemann surface is meant a proper, open, connected subset of a compact Riemann surface W whose boundary T is also the boundary of W— X and consists of a finite number of closed analytic arcs. Since W—X has an interior we may employ the Riemann-Roch Theorem to show that B(X) has enough functions to separate points and provide each point in X with a local uniformizer. Such a surface X therefore admits a natural homeomorphic imbedding into 9K(X) ; thus the corona conjecture is seen to be meaningful. Let X be a finite open Riemann surface. Ahlfors [ l ] has shown that there exists an analytic mapping po of X into the plane such that p = po\ 2f is an w-fold covering of X onto D and po(T) = D--D. Since r consists of closed analytic arcs, no ramification occurs on D — D. Clearly p*, the adjoint of p, is a C-isomorphism of B(D) into B(X), C being the complex field. Let B(D)* denote the range of p*, and for feB(D) let p*(f)=f*. Let cru denote the &th elementary symmetric function on n letters. For zÇzD let p~(z)= {xi(z)t • • • , xn(z)}9 each appearing to its multiplicity. Given fGB(X)9 <r*Cf(*i(*)). • • • ,ƒ(*»(*))) is in BCD). Thus, as is well known, B{X) is integrally dependent on B(D)*. Given NEW:(X) let M*~Nr\B(D)* and let PiN)**^*)-^*). Since 90?(X") and ffll(D) have the weak topology, P is continuous. Further, P is an extension of p. Since B{X) is integrally dependent on BCD)*, P is surjective. F o r / G ^ ( ^ ) ( ( ^ ( Z ) ) let ƒ denote the natural extension of ƒ to Wt(D)((m(X)). (See Hoffman [4, Chapter 10] for details.) Given /GJ3(D), / P = / * ^ . Let z denote the identity function