Ultrapower techniques in classical and noncommutative probability theory

In 1979 János Komlós proved a generalised result of the strong law of large numbers by removing the i.i.d condition, commonly known as the Komlós Theorem, which guarantees almost everywhere convergence of a subsequence of Cesàro averages for any L^1-bounded sequence of random variables. Recently, Junge, Scheckter, and Sukochev proved that a generalisation of the Komlós Theorem holds in noncommutative probability theory, where certain von Neumann algebras take the role of L^∞ spaces.

The proof relies on ultrafilters and ultrapowers, which originated in model theory but have found a wide range of usages throughout mathematics.

In this talk, we discuss the relationship between the classical and noncommutative probability theories as well as the application of ultrapower techniques in the framework of a proof of the classical Komlós Theorem, including an original result on the ultrapower of L^∞ spaces.