Real parts of uniform algebras

John M.F. O’Connell

Vol. 46 (1973), No. 1, 235–247

DOI: 10.2140/pjm.1973.46.235

Abstract

This paper is concerned with identifying those uniform algebras B on Γ = {z : |z| = 1} for which ReB—the space of real parts of the functions in B—equals ReA, where A denotes the disk algebra. It is shown that for any such algebra, there is an absolutely continuous homeomorphism Φ of Γ onto Γ so that B = A(Φ) = {f(Φ) : f ∈ A}. A partial converse to this theorem also holds: If Φ is a homeomorphism of Γ onto itself which is of class C² with nowhere vanishing derivative, then ReA(Φ) = ReA.

Mathematical Subject Classification 2000

Primary: 46J10

Milestones

Received: 23 July 1971

Published: 1 May 1973