可测算子对称空间若干几何性质研究

Symmetric space of measurable operators, which is known as non-commutative symmetric space, arises in a large number of applications in the fields of quantum statistical physics, quantum information and quantum computing. In this proposal, the geometric properties of symmetric space of measurable operators associated with an arbitrary semifinite von Neumann algebra with a faithful normal semifinite trace are studied by using the geometry theory of Banach space and the von Neumann algebra theory. Firstly, this project is to investigate monotone point, locally uniformly monotone point of the unit ball of symmetric spaces of measurable operators, and also to study the associated properties of uniform monotonicity, locally uniform monotonicity and strict monotonicity. Secondly, the existence and uniqueness of the best approximation element of the convex subset in the symmetric space, as well as the continuity of the best approximation operator are discussed by using the monotonicity. Finally, a concrete symmetric space of measurable operators is investigated, that is non-commutative Orlicz space. The formula or estimate of the monotone coefficients, the Opial modulus and the weakly convergent sequence coefficient of non-commutative Orlicz space is given by modular function. Furthermore, the existence condition of the fixed point property is presented for non-expansive mapping or multivalued non-expansive mapping in the non-commutative Orlicz Space.

可测算子对称空间，也称为非交换对称空间，在量子统计物理、量子信息与量子计算等领域有广泛应用。本项目拟借助Banach空间几何理论与von Neumman代数理论为研究工具，研究关于具有正规半有限忠实迹的半有限von Neumann代数的可测算子对称空间的几何结构。首先，讨论可测算子对称空间的单调点、局部一致单调点的判据，进一步给出该空间具有严格单调性、一致单调性判别条件；其次，利用单调性讨论可测算子对称空间凸子集的最佳逼近元的存在性、唯一性及最佳逼近算子的连续性；最后，讨论一类具体的可测算子对称空间——非交换Orlicz空间，通过模函数给出其单调系数、Opial模和弱收敛序列系数的计算公式或估计，进而寻求非交换Orlicz空间关于非扩张映射或集值非扩张映射具有不动点性质的条件。

非交换数学是与量子物理并驾齐驱的数学领域，它是研究量子统计物理、量子场论和量子信息与量子计算等许多物理学理论的数学基础。本项目研究属于非交换数学范畴。项目主要研究了具有正规半有限忠实迹的半有限von Neumann代数的可测算子对称空间关于单调性、Opial性质的继承性；并利用单调性来研究可测算子对称空间上子格、凸子集的最佳逼近问题，讨论了最佳逼近元的存在性、唯一性及最佳逼近算子的连续性的条件。其次，讨论了具体的可测算子对称空间——非交换Orlicz空间的最佳逼近元的具体刻画、单调性的等价条件及单调系数的计算。