ON THE ALMOST UNIFORM CONVERGENCE IN NONCOMMUTATWE L2-SPACES BY

Abstract. We introduce a kind of convergence in L, ovcr a von Neumann algebra and prove a few typical results being the analogues of classical pointwise theorems. I. In [4] a notion of the almost sure convergence in L, over a von Neumann algebra has been introduced and several limit theorems have been proved (cf. [6] ). The main goal of this paper * is to define another kind of convergence in the noncommutative L2-space which coincides with the ordinary almost everywhere convergence in the case of a commutative von Neumann algebra L, (X, 8, p). We shall call our new convergence the almost ungorm convergence in L,. Moreover, we prove some typical limit theorems (an individual ergodic theorem, a martingale convergence theorem, a Radema-cher-Menshov theorem) for this convergence. Let us begin with some notation and definitions. Let M be a a-finite von Neumann algebra with a faithful and normal state @. In our case, the GNS representation of (M, di) is faithful and normal so, without any loss o