【作者】 王佳；

【导师】 靳水林；

【作者基本信息】 哈尔滨工业大学 ， 基础数学， 2016， 硕士

【摘要】 在量子理论中,“两个可观测量是独立的”是很重要的术语,基于此,R.Haag在算子代数中引入了C*独立的定义,这样算子代数独立性成为算子代数领域重要的研究课题。近些年来,关于算子代数独立性的研究,许多学者通过不同的角度得到了诸多出色的成果。其中最重要的结果之一Roos定理表明C*代数A,B是C*独立的若A和B具有性质S。本文中,l是一个具有单位元e的C*代数,A和B是l的具有单位元的C*子代数。本文研究了算子代数的独立性。首先我们通过非耦合乘积态理论,描述了具有单位元的非交换的C*子代数的C*独立性:C*代数l的阿贝尔C*子代数A和C*子代数B是C*独立的当且仅当l具有A,B上的C*非耦合乘积性质;其次我们从态唯一扩张角度解决了具有单位元的非交换的C*子代数A和B的忠诚独立交换性问题;再者,我们定义了具有单位元的互相交换的C*子代数C*P-C独立并给出其等价刻画:A和B是C*P-C独立的当且仅当l与maxA?B同构。最后,我们在Banach空间引入了Hahn-Banach独立,同时证明了两个具有单位元的互相交换的C*子代数A和B是Hahn-Banach独立的当且仅当它们是C*独立的;并且给出了特殊情形下Hahn-Banach独立的等价条件。更多还原

【Abstract】 In quantum field theory, the term “two observables are independent” is of great significance. So in operator algebras, the concept of C* independence is introduced by R. Haag. And the independence of operator algebras is an important topic in operator algebra. In recent years, a lot of researchers obtain excellent works on the study of the independence of operator algebras. One of them is Roos Theorem and it indicates that twoC* algebras areC* independent if they satisfy the S property.In this paper, l is a C* algebra and has a unit e contained in subalgebra A and B.In the paper, we study the independence of operator algebras.First, we give the characterization of C* independence of twoC* subalgebras which have the unit and do not commute with each other through uncoupled product states: AbelianC* subalgebra A and C* subalgebra B that are not commuting with each other areC* independent if and only if l has theC* uncoupled product property;secondly, we solve the problem of faithfully independently commute of twoC* subalgebras which have the unit and do not commute with each other via unique common extensions of pairs of states; thirdly, we introduce the concept ofC*P-C independence of two C* subalgebras which have the unit and commute with each other and give the equivalent description of C*P-C independence: l is isomorphic to the maximal tensor product of twoC* subalgebras which commute with each other if and only if they are C*P-C independent.Last, we introduce Hahn-Banach independence,at the same time we proof that two C* subalgebras that commute with each other are Hahn-Banach independent if and only if they are C* independent Furthermore, we give the conditions of equivalence of C* algebras that are Hahn-Banach independent in a special case.更多还原