中心化子的刻画

Abstract

令X为实或复域F上的Banach空间,■为X上的标准算子代数,I是■的单位元.设Φ:■→■是可加映射.本文证明了,如果有正整数m,n,使得Φ满足条件Φ(A~(m+n+1))-A~mΦ(A)A~n∈FI对任意A成立,则存在λ∈F,使得对所有的A∈■,都有Φ(A)=λA.同样的结果对于自伴算子空间上的可加映射也成立.此外,本文还给出了中心素代数上满足条件(m+n)Φ(AB)-mAΦ(B)-nΦ(A)B∈FI的可加映射Φ的完全刻画.

Let X be a Banach space over the real or complex field F,let A be a standard operator algebra on X with unit I.Suppose thatΦ:A→A is an additive map and m,n are positive integers.It is proved that,ifΦsatisfiesΦ(A~(m+n+1)) - A~mΦ(A)A~n∈FI for all A∈A,then there exists someλ∈F such thatΦ(A) =λA for all A∈A.The same result is true for additive maps on the space of all self- adjoint operators.In addition,a complete characterization of mapsΦon centrally prime algebras satisfying (m+n)Φ(AB) - mAΦ( B ) - nΦ( A )B∈FI is obtained.