算子系统张量积的研究

In this project, we will do the research of the tensor product of operator systems, especially their local properties. We will give several nuclear properties of operator systems and study their hereditary properties. We also will compare these nuclear properties in the categories of C*-algebras and operator spaces though some examples. Finally, we will try to resolve the Connes embedding prblem in terms of operator systems. We will give some equivalent characterizations of Connes embedding problem in the category of operator systems. The main aim of this project is to solve some problems in operator algebras and discrete groups in terms of operator systems and operator spaces, which will make contributions in the mathematical branches of operator systems, operator spaces, group operator algebras and geometric group theory.

本项目研究算子系统张量积，着重考虑算子系统张量积的若干原子性质。我们将给出几种算子系统张量积的原子性的刻画及其等价刻画并且研究它们的一些遗传性质。我们还将通过一些实际例子来比较它们与C*代数和算子空间中的相应原子性的异同。最后，我们还希望通过算子系统来研究Connes embedding problem。我们会在算子系统范畴中给出Connes embedding problem的若干等价刻画，然后尝试证明这个问题或者给出反例。本项目的意义在于希望将算子系统和算子空间的技巧运用于解决算子代数和离散群的问题，对沟通算子系统、算子空间、群算子代数和几何群论这几个数学分支做出有价值的贡献。

本项目研究算子系统张量积，着重考虑算子系统张量积的若干原子性质。我们刻画了度量空间的相对性质A，算子空间的对偶性质及映射空间的局部性质，并将这些结果应用于算子系统。本项目的意义在于通过研究算子系统的局部性质，一方面能够解决离散群中的相关问题；另一方面也能用来解决量子信息和量子计算中的许多问题。