Mathematical Sciences: Noncommutative Harmonic Analysis and Operator Algebras

本文考虑了“非交换调和分析与算子代数”中的几个问题。该方案的框架主要是C*-代数。由希尔伯特空间上等距的一类半群生成的非自伴代数。利用非交换泊松变换、世界型分解、半群上的正定函数和等距膨胀等方法研究了这些代数及其表示。提议者期望找到一些经典结果的非交换类似物。本文拟进行的主要研究方向如下：1.数据分析；等距生成的C*代数上的泊松变换。2. 联合扩张与全称算子代数。3. 半群上的正定函数与等距表示。4. 单位圆盘上有界解析函数Hardy空间的非交换模拟。这项研究的动机是最近世界范围内对谐波分析的非交换方面的兴趣，起源于量子化的概念，它将数学的几个分支联系在一起，并与数学物理密切相关。本研究的目的是促进对这一相对较新的数学领域的理解，并将其中一些结果应用于非交换概率论、插值理论和模型理论。

9431954 POPESCU This proposal considers some problems in ``Noncommutative harmonic analysis and operator algebras''. The framework of this proposal is mainly the C*-algebra (resp. non-self-adjoint algebra) generated by a class of semigroups of isometries on Hilbert spaces. Noncommutative Poisson transforms, Wold-type decompositions, positive-definite functions on semigroups, and isometric dilations are considered to study these algebras and their representations. The proposer expects to find noncommutative analogs to some classical results. The main directions of this proposed research are the following: 1. Poisson transforms on some C*-algebras generated by isometries. 2. Joint dilations and universal operator algebras. 3. Positive-definite functions on semigroups and isometric representations. 4. Noncommutative analog of the Hardy space of bounded analytic functions on the unit disk. The motivation of this research is the recent worldwide interest in the noncommutative aspect of harmonic analysis originated from the concept of quantization which links together several branches of mathematics and is closely related to mathematical physics. The objective of this research is to advance the understanding of this relatively new area of mathematics and to apply some of these results to noncommutative probability theory, interpolation theory, and model theory.