Martingale Convergence Techniques in Noncommutative Integration

Abstract

The idea that the space of operators affiliated to a von Neumann algebra could be interpreted through the lens of integration theory stretches back to the original works of Murray and von Neumann on "rings of operators", the theory they developed to provide a mathematical framework for quantum mechanics. Since then, Segal discovered that not only is the theory analogous to integration, but that it is a genuine extension of Lebesgue integration theory. The key difficulty in understanding integration over von Neumann algebras is that there is no longer a notion of points or an underlying space. As such, we are driven to study extensions of classical problems in analysis, without access to the same techniques.

This thesis concerns how we may use novel techniques and constructions from the theory of martingales and Banach space geometry to solve problems in noncommutative analysis. The first problem we study is norm convergence for the Fourier transform of the noncommutative Vilenkin system. By using martingale constructions and techniques from noncommutative Calderon-Zygmund theory, we are able to prove a uniformly bounded weak type (1,1) estimate for the partial sums of the Fourier transform. This opens up classical problems from harmonic analysis in the noncommutative setting.

The second problem we consider is the extension of the Komlos theorem to general finite von Neumann algebras. The Komlos theorem resulted from a question of Steinhaus, and shows that given a uniformly norm bounded family of integrable functions, there exists a subset which "satisfies the strong law of large numbers", in that the Cesaro means converge almost everywhere to some fixed function. In proving this result, we resolve a long open question of Randrianantoanina, and introduce novel techniques for the study of almost everywhere convergence using ultrafilters, and the martingale structure of the infinitely iterated tower of ultrapowers.