Semi-Groups in L∞ And Local Ergodic Theorem

Abstract

AbstractWe show that any W*-continuous semi-group in L∞ is L1-norm continuous. As an application we prove the n-dimensional local ergodic theorem in L∞. We also note that any bounded additive process in L∞ is absolutely continuous.For n = 1 this local theorem improves those of R. Sato [14] and D. Feyel [6] and for n ≥ 1 it generalizes M. Lin's ones which hold for positive operators [12].