（1）本项目首先提出并致力于研究在算子空间中四类重要映射空间（完全原子映射空间、完全可积映射空间、完全1-可和映射空间和列Hilbert空间可分解映射空间）范畴中的内射性、局部自反性、正合性和原子性；.（2）第二个研究内容是探讨离散群G的性质与群C*-代数在各类映射空间范畴中局部性质之间的关系；.（3）Kirchberg在上世纪九十年代提出猜想：对C*-代数，正合性是否等价于局部自反性？至今没有实质性突破。本项目第三个目标是利用第一部分的研究成果来研究在四类映射空间范畴下的Kirchberg猜想；.（4）本项目的第四个研究内容是C*-代数在四类映射空间范畴下的LLP和WEP。由于算子代数中著名的QWEP猜想等价于对C*-代数，LLP=>WEP。我们将探索在新框架下的QWEP猜想。.项目申请人提出的映射空间范畴下的局部理论是算子空间中全新的研究方向，这方面的研究成果将是开创性且引人入胜。

(1) In this project we first introduce and investigate the injectivity, local reflexivity, exactness and nuclearity in the systems of four important mapping spaces (completely nuclear mapping spaces, completely integral mapping spaces, completely 1-summing mapping spaces and column Hilbert space factorable mapping spaces) in operator spaces;.(2) In the second part of this project we will investigate the relationship between the local properties of the discrete group G and its group C*-algebra;.(3) About twenty years ago, Kirchberg conjectured that the local reflexivity and exactness are equivalent for C*-algebras. By use of results in the first part, we will study this conjecture in these systems of mapping spaces;.(4) In the fourth part of this project, we want to study the local lifting property (LLP) and the weak expectation property (WEP) for C*-algebras in these systems of mapping spaces. Since Kirchberg’s another famous QWEP conjecture is equivalent to LLP=>WEP, we will investigate QWEP conjecture in these systems..Local theory of mapping spaces is a new research topic which is introduced by the applicant. Results in these areas will be pioneering and interesting.