Blaschke inductive limits of uniform algebras

Abstract

We consider and study Blaschke inductive limit algebrasA(b), defined as inductive limits of disc algebras A(D) linked by a sequence b={Bk}k=1∞ of finite Blaschke products. It is well known that big G-disc algebras AG over compact abelian groups G with ordered duals Γ=Gˆ⊂ℚ can be expressed as Blaschke inductive limit algebras. Any Blaschke inductive limit algebra A(b) is a maximal and Dirichlet uniform algebra. Its Shilov boundary ∂A(b) is a compact abelian group with dual group that is a subgroup of ℚ. It is shown that a big G-disc algebra AG over a group G with ordered dual Gˆ⊂ℝ is a Blaschke inductive limit algebra if and only if Gˆ⊂ℚ. The local structure of the maximal ideal space and the set of one-point Gleason parts of a Blaschke inductive limit algebra differ drastically from the ones of a big G-disc algebra. These differences are utilized to construct examples of Blaschke inductive limit algebras that are not big G-disc algebras. A necessary and sufficient condition for a Blaschke inductive limit algebra to be isometrically isomorphic to a big G-disc algebra is found. We consider also inductive limits H∞(I) of algebras H∞, linked by a sequence I={Ik}k=1∞ of inner functions, and prove a version of the corona theorem with estimates for it. The algebra H∞(I) generalizes the algebra of bounded hyper-analytic functions on an open big G-disc, introduced previously by Tonev.