A note on minimal envelopes of Douglas algebras, minimal support sets, and restricted Douglas algebras

Carroll Guillory

doi:10.1155/s0161171201006342

Abstract

We characterize the interpolating Blaschke products of finite type in terms of their support sets. We also give a sufficient condition on the restricted Douglas algebra of a support set that is invariant under the Bourgain map, and its minimal envelope is singly generated.

Highlights

Let H∞ be the Banach algebra of bounded analytic functions on the open unit disk D
We identify a function in H∞ with the Gelfand transform and consider H∞ the supremum norm closed subalgebra of the space of continuous functions on M(H∞)
By Carleson’s corona theorem, D is dense in M(H∞) in the weak∗-topology

Summary

Introduction

Let H∞ be the Banach algebra of bounded analytic functions on the open unit disk D. An interpolating Blaschke product b is said to be unimodular on trivial points if {x : |b(x)| < 1} ⊂ G. In [4], Hoffman proved that for x ∈ M(H∞), x ∈ G if and only if x ∈ Z(b) for some interpolating Blaschke product b. A part is called locally sparse if there is an interpolating Blaschke product b such that b(x) = 0 and |(b ◦ Lx) (0)| = 1.

Results

Conclusion