Closed subalgebras of the Banach algebra of continuously differentiable functions on an interval
type: Item
instance of: scholarly article
title: Closed subalgebras of the Banach algebra of continuously differentiable functions on an interval
zbMATH Open document ID: 0761.46046
author: John A. jun. Lindberg
author: Philip B. Downum
published in: Rocky Mountain Journal of Mathematics
publication date: 1992-06-25T00:00:00Z
review text: For \(f\in D^ 1(I)\), the Banach algebra of \(C^ 1\) complex functions on the closed interval \(I\), \(A_ f\) is the closed subalgebra generated by \(f\). When the range of \(f\) does not separate the plane, then \(g\in A_ f\) if and only if \(f(x)=f(y)\) implies both \(g(x)=g(y)\) and \(g'(x)f'(y)=g'(y)f'(x)\), and \(f'(x)=0\) implies \(g'(x)=0\). For real \(f\in D^ 1(I)\), an exhaustive set of conditions is given for \(A_ f\) to be a natural set defined in terms of point separation and zero sets of the derivatives. The authors establish various conditions for (Shilov) regularity of closed subalgebras (such as being inverse closed on \(I\)). The final section characterizes closed separating subalgebras over which \(D^ 1(I)\) is integral.
Mathematics Subject Classification ID: 46J30
zbMATH DE Number: 14136
zbMATH Keywords: differentiable functions
zbMATH Keywords: Shilov regularity of closed subalgebras
reviewed by: Edward J. Barbeau
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181072972
OpenAlex ID: W1964202128
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DOI: 10.1216/RMJM/1181072972
description: scientific article; zbMATH DE number 14136