Backward shifts on function algebras
type: Item
instance of: scholarly article
title: Backward shifts on function algebras
zbMATH Open document ID: 1217.47066
author: H. Takagi
author: H. Koshimizu
author: Hiroaki Ariizumi
published in: Journal of Mathematical Analysis and Applications
publication date: 2011-02-22T00:00:00Z
full work available at URL: https://soar-ir.repo.nii.ac.jp/
review text: In [Canad. Math. Bull. 31, 85--94 (1988; Zbl 0658.47032)], \textit{J. R. Holub} introduced the concept of backward shifts on Banach spaces. In [J. Math. Anal. Appl. 202, 485--491 (1996; Zbl 0878.47014)], \textit{M. Rajagopalan} and \textit{K. Sundaresan} proved that \(C(X)\), the Banach space of all continuous function on a compact Hausdorff space \(X\) equipped with the uniform norm, does not admit a backward shift if \(X\) is infinite. As a generalization, the authors in this paper show that an infinite-dimensional function algebra \(A\) on \(X\) which is a uniformly closed subalgebra of \(C(X)\) containing the constants and separating points of \(X\), does not admit a backward shift. Since a finite-dimensional space always admits a backward shift, the adjective ``infinite-dimensional'' is necessary here. Note that backward shifts on finite-dimensional space are not surjective, however, they are always surjective on infinite-dimensional spaces (see Proposition~1.2 in [\textit{M. Rajagopalan} and \textit{K. Sundaresan}, op.\,cit.]). Moreover, they define a backward quasi-shift as a weak type of a backward shift, and show that a function algebra \(A\) does not admit a surjective backward quasi-shift under the assumption that the Choquet boundary of \(A\) has at most finitely many isolated points. There are examples given in the end of the paper.
Mathematics Subject Classification ID: 47B37
Mathematics Subject Classification ID: 46J10
zbMATH DE Number: 5854864
zbMATH Keywords: backward shifts
zbMATH Keywords: function algebras
reviewed by: Ying-Fen Lin
MaRDI profile type: Publication
OpenAlex ID: W1985939082
cites work: On the separability problem for isometric shifts on \(C(X)\)
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DOI: 10.1016/J.JMAA.2010.10.012
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description: scientific article; zbMATH DE number 5854864