On individual subsequential ergodic theorem in von Neumann algebras

Abstract

We use a non-commutative generalization of the Banach Principle to show that the classical individual ergodic theorem for subsequences generated by means of uniform sequences can be extended to the von Neumann algebra setting. 0. Introduction. The study of almost everywhere convergence of the ergodic averages in the non-commutative setting was initiated by a number of authors among whom we mention Lance (5) and Yeadon (11). Individual ergodic theorems have been established for algebras with states as well as for algebras equipped with a seminite trace. The study of almost everywhere convergence of weighted and subsequential averages in von Neumann alge- bras is relatively new. So far, not much is known in this direction. Recently, a non-commutative analog of the classical Banach Principle, on convergence of sequences of measurable functions generated by a sequence of linear maps on L p -spaces, was established in (3). It is expected that, as in the commuta- tive case, this principle will be instrumental in obtaining various convergence results for the averages in non-commutative setting. In (8), an individual er- godic theorem for subsequences was proved, where the proof was based on application of the \commutative Banach Principle. In this paper we use the ergodic theorem of Yeadon (11) together with the results of (3), adjusted to the bilateral almost uniform convergence, to show that the main result of (8) also holds in the vNa setting. 1. Preliminaries. Let M be a von Neumann algebra (vNa) acting on a Hilbert space H. Let I be the unit of M, and let be a faithful normal seminite trace on M. Denote by P (M) the complete lattice of all projec- tions in M. Let A(M) be the set of all closed operators aliated with M. An operator x2 A(M) is said to be -measurable if for each > 0 there