An algebraic invariant for Jordan automorphisms on B(H):The set of idempotents

作     者：CUI Jianlian & HOU Jinchuan Department of Mathematical Science, Tsinghua University, Beijing 100084, China Department of Applied Mathematics, Taiyuan University of Technology, Taiyuan 030024, China Department of Mathematics, Shanxi Teachers University, Linfen 041004, China 

作者机构：Department of Mathematical Science Tsinghua University Beijing 100084 China Department of Applied Mathematics Taiyuan University of Technology Taiyuan 030024 China Department of Mathematics Shanxi Teachers University Linfen 041004 China 

出 版 物：《Science China Mathematics》 (中国科学：数学（英文版）)

年 卷 期：2005年第48卷第12期

页      面：1585-1596页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：supported by the National Natural Science Foundation of China(Grant No.10501029) Tsinghua Basic Research Foundation(to Cui) the National Natural Science Foundation of China(Grant No.1047 1082) a grant from PNSF of Shanxi(to Hou) 

主　　题：Hilbert space operators,Jordan automorphisms,idempotents. 

摘      要：Let H be an infinite dimensional complex Hilbert space. Denote by B(H) the algebra of all bounded linear operators on H, and by I(H) the set of all idempo-tents in B(H). Suppose that Φ is a surjective map from B(H) onto itself. If for every λ ∈ {-1,1,2,3,1/2,1/3} and A, B ∈ B(H), A - λB ∈ I(H) (?) Φ(A) - λΦ(B) ∈ I(H), then Φ is a Jordan ring automorphism, i.e. there exists a continuous invertible linear or conjugate linear operator T on H such that Φ(A) = TAT-1 for all A ∈ B(H), or Φ(A) = TA*T-1 for all A ∈ B(H); if, in addition, A-iB ∈ I(H) (?) Φ(A) -ιΦ(B) ∈ I(H), here ι is the imaginary unit, then Φ is either an automorphism or an anti-automorphism.