Β(H)上在恒等算子处可乘映射的特征

Characterization of multiplicative mappings at unit operator on B(H)

设A是代数,φ是A到自身的线性映射,如果对任意的S,T∈A且ST=Z,都有φ(ST)=φ(S)φ(T)成立,则称φ在Z处可乘.本文主要证明以下结果:设H是复数域上的无限维Hilbert空间,φ是Β(H)到自身强算子拓扑连续的线性满射,若φ在恒等算子I处可乘,则φ是空间自同构.

Let A be an algebra . We say that a linear mapping φ from A into itself is a muhiplicative mapping at Z（ Z ∈ A） if φ （ST） = φ （S） φ （T） for any S, T∈ A with ST = Z. Let H be an infinite dimensional complex Hilbert space and let φ be a surjective linear map on B （H） . In this paper, we prove that if φ is multiplicative at I and continuous in the stronger operator topology, then φ is an automorphism.