约化离散速降群交叉积C*-代数上的紧致量子度量空间结构

More and more attention has been paid to noncommutative metric geometry. It extends metric geometry to the realm of noncommutative geometry, and opens a new avenue for the study of metric geometry of noncommutative topological spaces and their applications to other fields, such as: fractal geometry, quantum field theory, string theory, high energy physics and mathematical physics. This project is devoted to the study of the compact quantum metric space structures on reduced crossed product C*-algebras of several important discrete groups (nilpotent groups, hyperbolic groups and rapid decay groups). The main goal of this project is to construct several new classes of compact quantum metric spaces from the resulting reduced crossed product C*-algebras by means of the equicontinuous actions of these groups on the compact quantum metric spaces induced by a spectral triple. This project will concentrate on the the compact quantum metric space structures of reduced crossed product C*-algebras in order to find new methods or thoughts to construct compact quantum metric spaces. This will promote the development of geometric group theory and noncommutative metric geometry, and reveal the much more effects of the properties of groups on operator algebras.

非交换度量几何的研究日益受到重视，它推广经典的度量几何到非交换几何中，以此来研究非交换拓扑空间上的度量几何，并将其应用到分形几何、量子场论、弦理论、高能物理和数学物理等其它自然科学领域。本项目将致力于研究几类重要的离散群（幂零群、双曲群和速降群）的约化交叉积C*-代数上的紧致量子度量空间结构问题。希望通过这些离散群等度连续地作用在由谱三元组诱导的紧致量子度量空间上得到它们的约化交叉积C*-代数上的紧致量子度量空间结构。本项目系统地研究约化交叉积C*-代数上的紧致量子度量空间结构，希望发现新的构造方法或思路。从而推动几何群论和非交换度量几何的发展，揭示群的性质对算子代数所产生的更加深刻的影响。

本项目主要研究几类离散群的约化交叉积C*-代数上的紧致量子度量空间结构。细致地刻画了紧致量子度量空间范畴，从多个角度描述了态射、单态射、满态射、等距态射和同构态射。进一步，对于群在紧致量子度量空间上的概周期型作用, 通过该空间的几个子集，该作用在对偶Banach空间的几个重要子集上的等度连续作用，以及Lip-范数和它在态空间上诱导的度量结构给出了几个刻画。我们引进了约化交叉积C*-代数上的序相容范数，并由此范数证明了离散速降群的约化交叉积C*-代数是一个紧致量子度量空间。本项目的研究对几何群论和非交换度量几何的发展有着一定的推动作用，揭示了群的性质对非交换紧致空间的度量结构产生的一些影响。