Blaschke inductive limits of uniform algebras

S. A. Grigoryan and T. V. Tonev

International Journal of Mathematics and Mathematical Sciences, 2001, vol. 27, 1-22

Abstract:

We consider and study Blaschke inductive limit algebras A ( b ) , defined as inductive limits of disc algebras A ( D ) linked by a sequence b = { B k } k = 1 ∞ of finite Blaschke products. It is well known that big G -disc algebras A G over compact abelian groups G with ordered duals Γ = G ˆ ⊂ 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 A G over a group G with ordered dual G ˆ ⊂ ℝ is a Blaschke inductive limit algebra if and only if G ˆ ⊂ 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 = { I k } 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.

Date: 2001
References: Add references at CitEc
Citations:

Downloads: (external link)
http://downloads.hindawi.com/journals/IJMMS/27/626754.pdf (application/pdf)
http://downloads.hindawi.com/journals/IJMMS/27/626754.xml (text/xml)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:626754

DOI: 10.1155/S0161171201006792

Access Statistics for this article

More articles in International Journal of Mathematics and Mathematical Sciences from Hindawi
Bibliographic data for series maintained by Mohamed Abdelhakeem ().