On Galois Correspondence and Non-Commutative Martingales

Details On Galois Correspondence and Non-Commutative Martingales On Galois Correspondence and Non-Commutative Martingales

2008-12-22

arxiv.org/abs/0812.3943v2

Mathematics

Operator Algebras

Scientific paper

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.

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