Blaschke inductive limits of uniform algebras

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.