International Journal of Mathematics and Mathematical Sciences
Volume 27 (2001), Issue 10, Pages 599-620
Blaschke inductive limits of uniform algebras
1Chebotarev Institute of Mathematics, Kazan State University, Kazan, Russia
2Department of Mathematical Sciences, University of Montana-Missoula, Missoula 59812-0864, MT, USA
Received 16 February 2001
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.
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.