Stochastic Banach Principle in Operator Algebras.pdf

a r X i v : m a t h / 0 6 0 6 4 8 7 v 1 [ m a t h .O A ] 2 0 J u n 2 0 0 6 STOCHASTIC BANACH PRINCIPLE IN OPERATOR ALGEBRAS GENADY YA. GRABARNIK AND LAURA SHWARTZ Abstract. Classical Banach principle is an essential tool for the investigation of the ergodic properties of C?esaro 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 subse- quences. 1. Introduction and Preliminaries In this paper we establish a Stochastic Banach Principle. The Banach Principle is one of the most useful tools in ”classical” point-wise ergodic theory. The Banach principle was used to give an alternative proof of the Birkhoff- Khinchin individual ergodic theorem. Typical applications of the Banach Principle are Sato’s theorem for uniform subsequences [17] and individual ergodic theorem for the Besicovitch Bounded sequences [15] . Non-commutative analogs for the (double side) almost everywhere convergence may be found in papers [8], [2]. In this paper we establish a Banach Principle for convergence in measure (Sto- chastic Banach Principle, Theorem 3.2.3). We reformulate the theorem in a form convenient for applications (Theorem 3.2.4). Based on the principle we give a simplified proof of the stochastic ergodic theorem (compare with [9]). We estab- lish stochastic convergence for Sato’s uniform subsequences (Theorem 3.6) and a stochastic ergodic theorem for the Besicovitch Bounded sequences (Theorem 3.7). Note that these results are new even in the commutative case. Throughout the paper we denote by M a von Neumann algebra with semi-finite normal faithful trace τ acting on Hilbert space H. Denote by P (M) the set of all orthogonal projections in M . Recall the following definitions (co