On the Uniform Zero-Two Law for Positive Contractions of Jordan Algebras

Abstract

Following an idea of Ornstein and Sucheston, Foguel proved the so-called uniform "zero-two" law: let \( T : L^1(X, \mathcal{F}, \mu) \to L^1(X, \mathcal{F}, \mu) \) be a positive contraction. If for some \( m \in \mathbb{N} \cup \{0\} \) one has \( \|T^{m+1} - T^m\| < 2 \), then \[ \lim_{n \to \infty} \|T^{n+1} - T^n\| = 0. \] In this paper we prove a non-associative version of the uniform "zero-two" law for positive contractions of \( L_1 \)-spaces associated with \( JBW \)-algebras.