A noncommutative generalisation of a problem of Steinhaus

Abstract

We extend the Révész and Komlós theorems to arbitrary finite von Neumann algebras, and in doing so solve an open problem of Randrianantoanina, removing the need for hyperfiniteness. The main result is the noncommutative Komlós theorem, which states that every norm-bounded sequence of operators in L1(M), for any finite von Neumann algebra M, admits a subsequence, such that for any further subsequence, the Cesàro averages converge bilaterally almost uniformly. This is a natural extension of Komlós' original result to the noncommutative setting.The necessary techniques which facilitate the proof also allow us to extend the Révész theorem to the noncommutative setting, which gives a similar subsequential law for series over bounded sequences in L2(M).