向量空间和C^(*)-代数上的扩张理论  

Dilation theory on vector spaces and C^(*)-algebras

作　　者：包琪瑶 韩德广 刘锐[1] BAO Qiyao;HAN Deguang;LIU Rui(School of Mathematical Sciences,Nankai University,300071,Tianjin,PRC;College of Sciences,Univrsity of Central Florida,Florida,USA)

机构地区：[1]南开大学数学科学学院,中国天津市300071 [2]中佛罗里达大学理学院,美国佛罗里达州

出　　处：《曲阜师范大学学报（自然科学版）》2022年第3期23-32,共10页Journal of Qufu Normal University(Natural Science)

基　　金：国家自然科学基金(12071230,11971348,11671214);南开大学百名青年学科带头人资助项目(63213027,91923104,91823003,63174012);中央高校基本科研业务费专项基金(63191503,63171225);美国国家自然科学基金(DMS-2105038).

摘　　要：著名的Naimark定理和Stinespring扩张定理表明每一个正算子值测度都有投影值扩张,作用于C^(*)-代数上的每一个完全有界线性映射都可以扩张为有界*-同态,这些都是Hilbert扩张.然而,在交换和非交换情形下,对于任意算子值测度和线性映射,都有基于Banach空间的一般扩张理论存在.这种一般的扩张理论最终可得到有界线性映射和算子值测度的分类理论.该文简要介绍含幺元代数和向量空间上代数版本的线性系统扩张理论,通过引进典则扩张和万有扩张两种自然的扩张结构,给出所有的线性极小同态扩张的主要分类结果;从Stinespring扩张出发,介绍C^(*)-代数上完全有界线性映射的刻画并说明即使对交换的纯原子的von Neumann代数也存在没有Hilbert扩张的例子.The celebrated Naimark and Stinespring’s dilation theorems show that every positive operator valued measure has projection valued dilation,and every completely bounded linear map acting on a C^(*)-algebra can be dilated to bounded*-homomorphism,all of which are Hilbertian dilations.However,a more general dilation theory exists for arbitrary operator valued measures and linear maps in both commutative and noncommutative settings based on Banach spaces.It is expected that such a general dilation theory may eventually lead to a classification theory for bounded linear maps and operator valued measures.This paper briefly introduces an algebraic version of the dilation theory for linear systems acting on unital algebras and vector spaces.With the introduction of two natural dilation structures,namely the canonical dilation and the universal dilation,the main results about the classifications of all linearly minimal homomorphism dilations are presented.Moreover,based on Stinespring’s dilation,the characterization of completely bounded maps acting on a C^(*)-algebra is introduced.Finally,an example to show the existence of non-Hilbertian dilation even for purely atomic abelian von-Neumann algebras is introduced.

关 键 词：线性系统 万有扩张 完全有界映射 C^(*)-代数 

分 类 号：O177.2[理学—数学]