Corona problem for Riemann surfaces of Parreau-Widom type
The author constructs Riemann surfaces of Parreau-Widom type for which the corona problem has a negative solution. Let R be a hyperbolic Riemann surface, \(H^{\infty}(R)\) the Banach algebra of all bounded (single- valued) analytic functions on R with the uniform norm and M(R) the maximal ideal space of \(H^{\infty}(R)\) with the Gelfand topology. For each \(z\in R\) let \(q_ z\) be the evaluation functional on \(H^{\infty}(R)\) defined by \(q_ z(f):=f(z)\). By setting \(\iota (z):=q_ z\), we get a canonical map sending R into M(R). The corona problem in the title asks if the image \(\iota\) (R) is dense in M(R). As is well known, the problem was originally posed by S. Kakutani and settled by L. Carleson in the unit disk. The problem for general Riemann surfaces has been solved negatively by B. Cole [cf. \textit{T. Gamelin}, Uniform algebras and Jensen measures (1978; Zbl 0418.46042)]. Let g(z,a) be the Green function for R with pole at a point \(a\in R\). For any \(a\in R\) and \(\alpha >0\) we denote by B(\(\alpha\),a) the first Betti number of the region \(\{\) \(z\in R:\) \(g(z,a)>\alpha \}\). Then the surface R is said to be of Parreau-Widom type if \(\int^{\infty}_{0}B(\alpha,a)d\alpha <\infty\) for some (and hence all) \(a\in R\). It is known that if R is of Parreau-Widom type the canonical map \(\iota\) is a homeomorphic embedding of R as an open subset of M(R). But the author gives an example of such surfaces for which \(\iota\) (R) is not dense in M(R). He constructs such a surface R by joining a sequence of so-called mulformed (finite-bordered) Riemann surfaces appearing in Cole's example by means of ''narrow'' bands. The key idea is that R becomes of Parreau-Widom type if the bands are sufficiently narrow.
scientific article; zbMATH DE number 3901124