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• W2035311033 abstract "We consider and study Blaschke inductive limit algebras <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$A (b)$ id=E1><mml:mrow><mml:mi>A</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>b</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>, defined as inductive limits of disc algebras<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$A(D)$ id=E2><mml:mrow><mml:mi>A</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>D</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>linked by a sequence<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$b ={B_k}^{infty}_{k = 1}$ id=E3><mml:mrow><mml:mi>b</mml:mi><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:msub><mml:mi>B</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>∞</mml:mi></mml:msubsup></mml:mrow></mml:math>of finite Blaschke products. It is well known that big<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$G$ id=E4><mml:mi>G</mml:mi></mml:math>-disc algebras<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$A_G$ id=E5><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mi>G</mml:mi></mml:msub></mml:mrow></mml:math>over compact abelian groups<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$G$ id=E6><mml:mi>G</mml:mi></mml:math>with ordered duals<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$Gamma = widehat{G} subset mathbb{Q}$ id=E7><mml:mi>Γ</mml:mi><mml:mo>=</mml:mo><mml:mover accent=true><mml:mi>G</mml:mi><mml:mo>ˆ</mml:mo></mml:mover><mml:mo>⊂</mml:mo><mml:mi>ℚ</mml:mi></mml:math>can be expressed as Blaschke inductive limit algebras. Any Blaschke inductive limit algebra<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$A(b)$ id=E8><mml:mi>A</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>b</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>is a maximal and Dirichlet uniform algebra. Its Shilov boundary<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$partial A(b)$ id=E9><mml:mo>∂</mml:mo><mml:mi>A</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>b</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>is a compact abelian group with dual group that is a subgroup of<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$mathbb{Q}$ id=E10><mml:mi>ℚ</mml:mi></mml:math>. It is shown that a big<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$G$ id=E11><mml:mi>G</mml:mi></mml:math>-disc algebra<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$A_G$ id=E12><mml:msub><mml:mi>A</mml:mi><mml:mi>G</mml:mi></mml:msub></mml:math>over a group<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$G$ id=E13><mml:mi>G</mml:mi></mml:math>with ordered dual<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$widehat{G}subsetmathbb{R}$ id=E14><mml:mover accent=true><mml:mi>G</mml:mi><mml:mo stretchy=true>ˆ</mml:mo></mml:mover><mml:mo>⊂</mml:mo><mml:mi>ℝ</mml:mi></mml:math>is a Blaschke inductive limit algebra if and only if<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$widehat{G}subsetmathbb{Q}$ id=E15><mml:mover accent=true><mml:mi>G</mml:mi><mml:mo stretchy=true>ˆ</mml:mo></mml:mover><mml:mo>⊂</mml:mo><mml:mi>ℚ</mml:mi></mml:math>. 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<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$G$ id=E16><mml:mi>G</mml:mi></mml:math>-disc algebra. These differences are utilized to construct examples of Blaschke inductive limit algebras that are not big<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$G$ id=E17><mml:mi>G</mml:mi></mml:math>-disc algebras. A necessary and sufficient condition for a Blaschke inductive limit algebra to be isometrically isomorphic to a big<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$G$ id=E18><mml:mi>G</mml:mi></mml:math>-disc algebra is found. We consider also inductive limits<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$H^{infty}(I)$ id=E19><mml:msup><mml:mi>H</mml:mi><mml:mi>∞</mml:mi></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>I</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>of algebras<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$H^{infty}$ id=E20><mml:msup><mml:mi>H</mml:mi><mml:mi>∞</mml:mi></mml:msup></mml:math>, linked by a sequence<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$I = {I_k}^{infty}_{k = 1}$ id=E21><mml:mi>I</mml:mi><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:msub><mml:mi>I</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow><mml:mo>}</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>∞</mml:mi></mml:msubsup></mml:math>of inner functions, and prove a version of the corona theorem with estimates for it. The algebra<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$H^{infty}(I)$ id=E22><mml:msup><mml:mi>H</mml:mi><mml:mi>∞</mml:mi></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>I</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>generalizes the algebra of bounded hyper-analytic functions on an open big<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$G$ id=E23><mml:mi>G</mml:mi></mml:math>-disc, introduced previously by Tonev." @default.
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• W2035311033 title "Blaschke inductive limits of uniform algebras" @default.
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