On Galois Correspondence and Non-Commutative Martingales

TL;DR

This thesis explores Galois correspondence in von Neumann algebras and develops a non-commutative probability framework, establishing correspondences between group actions and subalgebras, and analyzing non-abelian martingales.

Contribution

It introduces a non-commutative probability theory for von Neumann algebras, extending Galois correspondence to non-abelian group actions and martingale convergence.

Findings

Established one-to-one correspondences between subgroups and subalgebras for various automorphism groups

Identified and characterized non-abelian martingales in the non-commutative setting

Proved convergence theorem for non-abelian martingales

Abstract

The subject of this thesis is Galois correspondence for von Neumann algebras and its interplay with non-commutative probability theory. After a brief introduction to representation theory for compact groups, in particular to Peter-Weyl theorem, and to operator algebras, including von Neumann algebras, automorphism groups, crossed products and decomposition theory, we formulate first steps of a non-commutative version of probability theory and introduce non-abelian analogues of stochastic processes and martingales. The central objects are a von Neumann algebra $\Ma$ and a compact group $\Gr$ acting on $\Ma$, for which we give in three consecutive steps, i.e. for inner, spatial and general automorphism groups one-to-one correspondences between subgroups of $\Gr$ and von Neumann subalgebras of $\Ma$. Furthermore, we identify non-abelian martingales in our approach and prove for them a convergence theorem.