Harmonic Analysis on Operator Algebras

对于Cuntz-Pimsner代数的规范作用量出现量子群作用和Kms态的各种模型，我确定了(非对易)泊松边界。特别是，我确定了Connes和Takesaki为几个型号引入的重量流。由于权流的定义涉及遍历分解，因此不容易给出具体的一般描述。Hilbert空间H的有界算子集B(H)是von Neumann代数。B(H)的单参数单位保持自同态半群称为E_0-半群。E_0-半群分为三类，即类型I、类型II和类型III，除类型I外，E_0-半群的结构还不是很清楚。在R·斯里尼瓦桑的帮助下，我构造了无数个相互非上循环共轭的III型E_O-半群。在我们构造之前，唯一已知的例子是由Tsirelson构造的。由于Tsirelson的不变量对于我们的例子来说是微不足道的，他的方法不能区分我们的例子。对于给定的E_0-半群和单位区间[0，1]的开子集U，可以结合一个von Neumann代数A(U)，它是E_0-半群的上循环共轭不变量。Murray和von Numann将von Neumann代数分为I型、II型和III型三类，这与先验的E_0-半群的类型分类无关。对于I型和II型E_0-半群，von Neumann代数A(U)总是I型的。我们证明了，对于我们的例子，根据集合U的形状，Von Neumann代数A(U)可以是III型的。

For various models appearing quantum group actions and KMS states for the gauge action of the Cuntz-Pimsner algebra, I determined the (non-commutative) Poisson boundaries. In particular, I determined the flow of weights introduced by Connes and Takesaki for several models. Since the definition of the flow of weights involves ergodic decomposition, it is not so easy to give a concrete description in general.The set of bounded operators B(H) of a Hilbert space H is a von Neumann algebra. A one-parameter semigroup of unit preserving endomorphism of B(H) is said to be an E_0-semigroup. E_0-semigroups are classified into three categories, type I, type II, and type III, and except for the type I case, the structure of E_0-semigroups is not well-understood. With R. Srinivasan, I constructed uncountably many mutually non-cocycle conjugate E_O-semigroups of type III. Before our construction, the only known such examples were constructed by Tsirelson. Since Tsirelson's invariant is trivial for our examples, his method can not distinguish our examples. For a given E_0-semigroup and for an open subset U of the unit interval [0,1], one can associate a von Neumann algebra A(U), which is a cocycle conjugacy invariant of the E_0-semigroup. Murray and von Numann classified von Neumann algebras into three categories, type I, type II, and type III, which is nothing to do with the type classification of E_0-semigroups a priori. For type I and type II E_0-semigroups, the von Neumann algebra A(U) is always of type I. We show that for our examples, the von Neumann algebra A(U) may be of type III according to the shape of the set U.