关于Galois对应与非交换鞅
On Galois Correspondence and Non-Commutative Martingales
摘要：
本论文的主题是冯诺依曼代数的伽罗瓦对应及其与非交换概率论的相互作用。在简要介绍了紧群的表示论，特别是Peter-Weyl定理，算子代数，包括冯诺依曼代数，自同构群，交叉产品和分解理论，我们制定了第一个步骤的非交换版本的概率论，并介绍非阿贝尔类似的随机过程和鞅。中心对象是von Neumann代数$\Ma$和作用在$\Ma$上的紧群$\Gr$，对于它们，我们分三步给出了$\Gr$的子群与$\Ma$的von Neumann子代数之间的一一对应关系，即对于内部、空间和一般自同构群.此外，我们确定非交换鞅在我们的方法，并证明了他们的收敛定理。
英文摘要：
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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