  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.
