Shift-invariant algebras and inductive limit algebras on groups

Abstract

Uniform algebras that can be expressed as inductive limits of standard simpler algebras are of particular interest. For instance, some G-disc algebras are inductive limits of sequences of disc algebras, connected with finite Blaschke products, called also Blaschke inductive limit algebras. Here we show, among others, that only G-disc algebras, and the spaces $H^\infty (\mathbb{D}_G )$ with G ⊂ ℚ, can be expressed as limits of countable inductive sequences of algebras of type $A(\mathbb{D})$ and H^∞ correspondingly. We study also inductive limits of sequences of spaces of type H^∞ and prove corresponding corona theorems. Further, we establish relationships between Bourgain algebras of coordinate algebras, and the Bourgain algebra of their inductive limit, and also between H^∞-spaces on $\mathbb{D}$, as coordinate algebras in an inductive sequence, and their inductive limit. While we state all results for general shift-invariant algebras A_S, they apply automatically to the particular cases of algebras AP_S of almost periodic functions, and of H ^∞ _S -algebras.