交叉积是构造新算子代数最为重要的方法之一，也是算子代数理论中的重要研究对象。目前，C*-代数交叉积理论已经较为成熟，但非自伴算子代数交叉积的研究仍方兴未艾。Katsoulis和Ramsey于2015年引入了并研究了非自伴算子代数的交叉积。对于这种一般算子代数交叉积的研究仍处于初始阶段，有诸多问题还有待解决。. 本项目以研究非自伴算子代数交叉积的结构为中心课题。具体来说，我们着力于刻画局部紧群的Imai-Takai对偶性、imprimitivity性质、Hao-Ng同构等性质。为了达成目标，我们需要定义局部紧群的余作用，刻画导出代数的C*-覆盖和C*-包络，揭示导出代数的C*-包络与C*-包络的导出代数之间的关系。此外，交叉积的C*-包络和极大C*-覆盖、以及交叉积算子代数的C*-包络与C*-包络的交叉积之间的同构性也是我们计划研究的重要问题。

Crossed product is one of the most important methods to construct new operator algebras and plays a central role in the theory of operator algebras. Although the theory of crossed product C*-algebras has reached a certain maturity, the study of crossed product non-selfadjoint operator algebras is far from completion. In 2015, Katsoulis and Ramsey introduced and studied the crossed products of non-selfadjoint operator algebras. Their theory of crossed products of arbitrary operator algebras is not well understood and many questions are still waiting to be answered. . The major goal in our proposed research is to understand the structure of crossed products non-selfadjoint operator algebras. More specifically, we are interested in the Imai-Takai duality theorem, the imprimitivity property, and the Hao-Ng isomorphism problem for a locally compact group. In order to achieve this research goal, we will need to define the coation of a locally compact group, characterize the C*-covers and C*-envelopes of the induced algebras and uncover the relationship between the C*-envelopes of the induced algebras and the induced algebras of the C*-envelopes. We will also study the C*-envelopes and the maximal C*-covers of the crossed product operator algebras, and check whether the C*-envelope of the crossed product operator algebra is isomorphic to the crossed product of the C*-envelope.