The zero-two law in Banach lattice algebras
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.
scientific article; zbMATH DE number 4105608