The zero-two law in Banach lattice algebras
type: Item
instance of: scholarly article
title: The zero-two law in Banach lattice algebras
zbMATH Open document ID: 0675.46013
DOI: 10.1007/BF02767367
published in: Israel Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: The zero-two law in \(L_ p\)-spaces and, more generaly, in Banach lattices, has been studied by many authors recently (e.g. Zaharopol, Wittmann, Katznelson and Tzafriri, Schaefer and Schep). In the present paper the author shows that Schep's result holds in an arbitrary unital Banach lattice algebra, to the effect that if A is a unital Banach lattice algebra and \(a\in A^+\) satisfies \(\| a\| \leq 1\), then either \(\| a^{n+1}-a^ n\| =2\), or else \(\| a^{n+1}-a^ n\| \to 0\) as \(n\to \infty.\) The proof is self-contained and does not appeal to earlier results. Recent related versions of the zero-two law in unital Banach algebras (and the connections with the classical Gelfand-result) can be found in: (i) \textit{J. Esterle}, Lect. Notes Math. 975, 66-162 (1983; Zbl 0536.46041), theorem 9.1. (iii) \textit{G. R. Allan}, \textit{T. J. Ransford}, Power dominated elements in a Banach algebra, Stud. Math. 94, 63-79 (1989), theorems 1.1 and 1.2.
Mathematics Subject Classification ID: 46B42
Mathematics Subject Classification ID: 47B60
Mathematics Subject Classification ID: 46H05
zbMATH DE Number: 4105608
zbMATH Keywords: zero-two law in Banach lattice algebras
zbMATH Keywords: f-algebra
zbMATH Keywords: peripheral spectrum
zbMATH Keywords: cyclic
zbMATH Keywords: zero-two law in unital Banach algebras
author: Jacobus J. Grobler
reviewed by: Charles B. Huijsmans
MaRDI profile type: Publication
cites work: Q3215824
cites work: On power bounded operators
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cites work: The modulus of a regular linear operator and the ''zero-two'' law in \(L^ p\)-spaces \((1<p<+\infty,p\neq 2)\)
full work available at URL: https://doi.org/10.1007/bf02767367
OpenAlex ID: W2091763364
description: scientific article; zbMATH DE number 4105608