International Journal of Mathematics and Mathematical Sciences
Volume 27 (2001), Issue 10, Pages 599-620

Blaschke inductive limits of uniform algebras

S. A. Grigoryan1 and T. V. Tonev2

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 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 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 . It is shown that a big G-disc algebra AG over a group G with ordered dual Gˆ is a Blaschke inductive limit algebra if and only if Gˆ. 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.