Some conditions on Douglas algebras that imply the invariance of the minimal envelope map

Carroll Guillory

doi:10.1155/s0161171201004148

Abstract

We give general conditions on certain families of Douglas algebras that imply that the minimal envelope of the given algebra is the algebra itself. We also prove that the minimal envelope of the intersection of two Douglas algebras is the intersection of their minimal envelope.

Highlights

Let D denote the open unit disk in the complex plane, and T the unit circle
Let Ethin = [H∞ : q | q is a thin interpolating Blaschke product], Efin = [H∞ : b | b is of finite type G], and EG = [H∞ : φ | φ is of type G]
What conditions must be imposed on the Douglas algebra B such that BW = [BW ]m? For example, if B = H∞[q], where q is an interpolating Blaschke product of type G, and W is a weak peak set for B, is BW = [BW ]m?

Summary

Introduction

Let D denote the open unit disk in the complex plane, and T the unit circle. By L∞ we mean the space of essential bounded measurable functions on T with respect to the normalized Lebesgue measurement. A point x ∈ M(H∞) is called locally thin if there is an interpolating Blaschke product q such that q(x) = 0 and 1 − Zn(α) 2 q Zn(α) → 1 (1.11) Let Ethin = [H∞ : q | q is a thin interpolating Blaschke product], Efin = [H∞ : b | b is of finite type G], and EG = [H∞ : φ | φ is of type G].

Results

Conclusion