Weak-star closed algebras containing R(K)
Let \(K\subset {\mathbb{C}}\) be a compact subset of the complex plane, and let R(K) be the function algebra consisting of the uniform limits on K of rational functions with poles off K. Let \(\nu\) be a finite positive measure on the boundary of K, and let \(R^{\infty}(\nu)\) be the weak- star closure of R(K) in \(L^{\infty}(\nu)\). This paper deals with the problem of classifying the weak-star closed subalgebras of \(L^{\infty}(\nu)\) containing \(R^{\infty}(\nu)\). When K is the unit disk and \(\nu\) is the arc-length measure on the boundary of K, then \(R^{\infty}(\nu)=H^{\infty}(d\theta)\) is a maximal weak-star closed subalgebra of \(L^{\infty}(\nu)\). There is a similar weak-star maximality theorem in the case K is finitely connected and has connected interior and \(\nu\) is a harmonic measure on the boundary of K. The author deals with the case of infinitely connected sets K such that (1) the interior of K is connected and dense in K, (2) there is a free boundary arc in the boundary of K, (3) R(K) is pointwise boundedly dense in \(H^{\infty}(K^ 0).\) The main result, which describes the proper weak-star closed subalgebras of \(L^{\infty}(\nu)\) containing \(R^{\infty}(\nu)\), is then applied to the roadrunner set and the champagne bubble set.
classifying the weak-star closed subalgebras of \(L^{\infty }(\nu )\) containing \(R^{\infty }(\nu )\)
scientific article; zbMATH DE number 4109385