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 AS, they apply automatically to the particular cases of algebras APS of almost periodic functions, and of H S ∞ -algebras.