An algebraic invariant for Jordan automorphisms on B(H): The set of 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 idempotents 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) = T A~*T~(-1) for all A ∈ B(H); if, in addition, A - iB ? I(H) < = > Φ(A) - iΦ(B) ∈ I(H), here i is the imaginary unit, then Φ is either an automorphism or an anti-automorphism.