非交换Orlicz空间的性质及其闭子空间

Noncommutative Orlicz space is the focus of operator algebra in recent years, in this project, we want to study some problems which are closely related with the properties of noncommutative Orlicz space and its closed spaces. On the one hand, we will study the properties of noncommutative Orlicz spaces. We first study the basic properties of operators in noncommutative Orlicz spaces. Moreover, the topological properties of noncommutative Orlicz spaces are studied by generalizing the topologies on B(H). At last, we generalize the measurable operators and study the properties of locally measurable operators. On the other hand, we study a special kind of noncommutative Orlicz space, Hilbert space, and find the condition that the Orlicz function φ needs to satisfy, so that L_φ(M,τ) is a Hilbert space. Then we also consider the index of nocommutative Orlicz spaces L_φ(M,τ) generated by the factor M of type II_1 to closed subspace L_φ(M,τ) generated by the subfactor N of M..Based on the spectral decomposition and unbounded operator theory, this project studies the properties and closed subspaces of noncommutative Orlicz space from the perspective of pure algebra, thus improving the noncommutative Orlicz space theory.

非交换Orlicz空间是近年来算子代数的研究热点,本项目拟围绕非交换Orlicz空间的性质和闭子空间进行研究.一方面,研究非交换Orlicz空间的性质:首先,研究非交换Orlicz空间中算子的性质;其次,通过推广B(H)上的拓扑,研究非交换Orlicz空间的拓扑性质;最后,推广可测算子,研究局部可测算子的性质.另一方面,研究一类特殊的非交换Orlicz空间——Hilbert空间,找到Orlicz函数φ需要满足的条件,使得L_φ(M,τ)是Hilbert空间.并进一步考虑II_1型因子M生成的非交换Orlicz空间L_φ(M,τ)到由M的子因子N生成的闭子空间L_φ(M,τ)的指标..本项目就是根据算子的谱分解和无界算子理论,从纯代数角度研究非交换Orlicz空间的性质及其闭子空间,从而完善非交换Orlicz空间理论.

本项目基于无界算子的谱分解和投影算子的性质，从纯代数角度对非交换Orlicz空间中算子,闭子空间的性质等进行研究。首先，进一步讨论了可测算子的收敛性，此部分内容是对已有可测算子收敛性相关结论的推广，丰富了可测算子空间的研究成果，并为非交换Orlicz空间中算子的研究提供一定思路和理论基础。其次，讨论了非交换Orlicz空间中算子依范数收敛、依测度收敛、依Orlicz范数收敛之间的关系。最后，研究了非交换Orlicz空间的闭子空间的性质，证明了L_φ (M,τ)的子空间E_φ (M,τ)和A_φ (M,τ)的等价性。相关研究将加深对非交换Orlicz空间的理解,同时本项目的研究成果丰富和完善了非交换 L_p空间理论，为这一空间提供新的研究思路。