A Noncommutative Generalisation of a Problem of Steinhaus

Abstract:

Responding to work of Révész on a problem of Steinhaus, Komlós showed that every integrable sequence of random variables contains a subsequence which “satisfies the strong law of large numbers”, such that the sequence of Cesàro averages converges almost everywhere. We solve an open problem of Randrianantoanina by extending this result to arbitrary semifinite von Neumann algebras. In this talk we discuss the history of the problem, what it means to study almost everywhere convergence in the noncommutative setting, the technical obstructions of the noncommutative setting, and several related results.