S. A. Grigoryan, T. V. Tonev, "Blaschke inductive limits of uniform algebras", International Journal of Mathematics and Mathematical Sciences, vol. 27, Article ID 626754, 22 pages, 2001. https://doi.org/10.1155/S0161171201006792
Blaschke inductive limits of uniform algebras
We consider and study Blaschke inductive limit algebras, defined as inductive limits of disc algebras linked by a sequence of finite Blaschke products. It is well known that big -disc algebras over compact abelian groups with ordered duals can be expressed as Blaschke inductive limit algebras. Any Blaschke inductive limit algebra is a maximal and Dirichlet uniform algebra. Its Shilov boundary is a compact abelian group with dual group that is a subgroup of . It is shown that a big -disc algebra over a group with ordered dual is a Blaschke inductive limit algebra if and only if . 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 -disc algebra. These differences are utilized to construct examples of Blaschke inductive limit algebras that are not big -disc algebras. A necessary and sufficient condition for a Blaschke inductive limit algebra to be isometrically isomorphic to a big -disc algebra is found. We consider also inductive limits of algebras , linked by a sequence of inner functions, and prove a version of the corona theorem with estimates for it. The algebra generalizes the algebra of bounded hyper-analytic functions on an open big -disc, introduced previously by Tonev.
Copyright © 2001 Hindawi Publishing Corporation. 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.