Mathematical Sciences: Noncommutative Harmonic Analysis and Operator Algebras

小行星9431954 这个建议考虑了“非交换调和分析与算子代数”中的一些问题。这个建议的框架主要是C*-代数（分别为）。非自伴代数）生成的一类半群的等距Hilbert空间。非交换Poisson变换，Wold型分解，半群上的正定函数和等距扩张被认为是研究这些代数及其表示。提出者期望找到一些经典结果的非交换类似物。这项拟议研究的主要方向如下： 1.等距生成的C ~*-代数上的Poisson变换 2.联合扩张与泛算子代数。 3.半群上的正定函数与等距表示。 4.上有界解析函数的哈代空间的非交换模拟 单位磁盘。 这项研究的动机是最近世界范围内的兴趣，在非交换方面的谐波分析起源于量子化的概念，它连接在一起的几个分支的数学，并密切相关的数学物理。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.