S. A. Grigoryan¹ and T. V. Tonev²

Copyright © 2001 S. A. Grigoryan and T. V. Tonev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.