Stochastic Banach principle in operator algebras

Abstract

Classical Banach principle is an essential tool for the investigation of the ergodic properties of Česaro subsequences.The aim of this work is to extend Banach principle to the case of the stochastic convergence in the operator algebras.We start by establishing a sufficient condition for the stochastic convergence (stochastic Banach principle). Then we formulate stochastic convergence for the bounded Besicovitch sequences, and, as consequence for uniform subsequences.

The Banach Principle for ideal convergence in the classical and noncommutative context

“…each T n continuous, the whole family pointwise uniformly bounded almost everywhere), one can actually obtain an estimate on the 'rate of continuity' [4], as expressed by the condition (3). In the noncommutative framework analogous conditions were considered in [6]. Can any estimate of that type be obtained for the general I-case?…”

“…In recent years there has been a lot of work on the noncommutative generalisations of the Banach Principle, where a.e. convergence of a family of measurable functions is replaced by almost uniform (or bilaterally almost uniform, or stochastic) convergence of measurable operators affiliated with a given von Neumann algebra [2,3,5,6,14]. The aim of this paper is to establish both classical and noncommutative versions of the Banach Principle for various notions of convergence related to an ideal.…”