20世纪60年代Arveson提出的非交换Hardy空间的理论被重视并且有了长足发展。交叉积作为算子理论的基本工具之一，诸多学者将其与非交换理论结合起来研究并取得了很多重要成果。本项目将继续对非交换理论和交叉积进行深入研究。令L和L+分别是冯·诺依曼代数M上的W*交叉积和解析交叉积。1）我们讨论由K和L2(M)生成的向量值空间的Laurent算子集与L的等价性，其中K是整数或者实数集。并讨论附属于L+的非交换Hardy空间和BMO空间的性质。特别地，给出L+下的非交换H1-BMO对偶。2）在附属于L+的非交换Hardy空间上定义Toeplitz和Hankel算子。分别讨论Toeplitz算子的谱包含性质和Hankel算子的Nehari问题，以及给出Hankel算子与BMO间的关系。3）讨论相伴于W*交叉积的BMO空间的插值。

The theory of the noncommutative Hardy space has been paid much attention and developed since it was proposed by Arveson in the 1960s. The crossed product, as one of basic tools of the operator theory, is studied by many scholars in combination with the noncommutative theory and lots of important research achievements have been obtained. This project will make an in-depth study on the noncommutative theory and the crossed product. Let L and L+ be a W*-crossed product and an analytic crossed product of the von Neumann algebra M, respectively. 1) We discuss the equivalence between the crossed product L and the set of Laurent operators on a vector value Lesbegue space generated by L2(M) and K, where K is the set of integers or real numbers. And we study the fundamental properties of the noncommutative Hardy spaces and the noncommutative BMO spaces associated with the analytic crossed product L+, especially the noncommutative H1-BMO duality. 2) We define the Toeplitz and Hankel operators on the noncommutative Hardy spaces. We study the spectrum of Toeplitz operators and the Nehari's problem of Hankel operators, and discuss the equivalence between the norms of Hankel operators and the BMO norms of its symbols. 3) We study the interpolation of noncommutative BMO spaces associated with the W*-crossed products.