非交换Hardy空间上的算子代数研究

In this proposal, we aim to thoroughly understand the operator algebras on a noncommutative Hardy space based on a subdiagonal algebra in a general von Neumann algebra. We will apply the modular theory of von Neumann algebras to noncommutative Hardy space and consider the group action on invariant subspaces of subdiagonal algebras by the induced automorphisms of modular automorphism groups on noncommutative Lp spaces. The intrinsic structure of invariant subspaces for subdiagonal algebras will be obtained. We will also establish a correspondence between invariant subspaces and weak* closed submodules of subdiagonal algebras in von Neumann algebras and focus on the structure of those subdiagonal algebras for which any invariant subspace in the noncommutative Hardy space H2 has Beurling type. From these basis, we will study the reflexivity, hereditary reflexivity as well as algebraic commutants of the noncommutative analytic Toeplitz algebras. Furthermore, using the intrinsic structure of invariant subspaces, we aim to investigate the analytic characterizations of the subalgebras containing a subdiagonal algebra in a von Neumann algebra. We then will characterize the features when a reflexive subalgebra in a von Neumann algebra becomes a subdiagnal algebra and obtain generators in the relative invariant subspace lattices of the subdiagonal algebra.

基于一般von Neumann代数上次对角代数建立的非交换Hardy空间，本课题拟对非交换Hardy空间上的算子代数做深层次分析。应用von Neumann代数的模理论，通过模同构群在非交换Lp空间诱导的自同构，考察模同构群在次对角代数的不变子空间上的群作用，获得次对角代数的不变子空间的内在结构，并建立不变子空间与von Neumann代数中次对角代数的弱*闭子模的对应，聚焦一类在非交换Hardy空间H2上具有Beurling型不变子空间的次对角代数的结构分析。在此基础上，研究非交换Hardy空间上的解析Toeplitz代数的自反性、遗传自反性以及代数换位等问题。进一步地，我们拟应用不变子空间的内在结构，分析von Neumann代数中包含次对角代数的子代数解析特征，刻画von Neumann代数中自反子代数成为次对角代数的特征，进而获得次对角代数的相对不变子空间格的生成元。

四年来，课题组在非交换Hardy空间上的算子代数以及相关问题等方面展开了系统深入的研究，完成了以下研究工作. 1. 我们定义了1型次对角代数，得到了其生成元结构. 完全给出了p=1,2,∞时Lp (M)中的不变子空间L构造，建立了Lp (M)中的不变子空间与M中W*闭子模的一一对应关系；2. 在1型非交换Hp空间上，证明了基于1重次对角代数的解析Toeplitz代数的遗传自反性，解决了1型非交换Hp空间上的解析Toeplitz算子代数的代数换位问题；3. 给出了von Neumann代数中1型次对角代数的代数极大性的充分必要条件；4. 在1型次对角代数中，肯定回答了Arveson关于有限von Neumann代数的次对角代数是否自动有限的问题. 与此同时 我们也研究了算子代数上的序结构、正交投影对以及保持问题.