Interpolation in strongly logmodular algebras

Rahman Mahmoud Younis

Vol. 102 (1982), No. 1, 247–251

DOI: 10.2140/pjm.1982.102.247

Abstract

Let A be a strongly logmodular subalgebra of C(X), where X is a totally disconnected compact Hausdorff space. For s a weak peak set for A, define A_s = {f ∈ C(X) : f|_s ∈ A|_s}. We prove the following:

Theorem1. Let s be a weak peak set for A. If b is an inner function such that b|_s is invertible in A|_s then there exists a function F in A ∩ C(X)⁻¹ such that F = b on s.

Theorem 2. Let s be a weak peak set for A. If U ∈ C(X), |U| = 1 on s and dist(U,A_s) < 1, then there exists a unimodular function Ũ in C(X) such that Ũ = U on s and dist(Ũ,A) < 1.

Mathematical Subject Classification 2000

Primary: 46J10

Milestones

Received: 5 May 1980

Published: 1 September 1982