算子代数上保交换零积的可加映射

Commutative zero-product preserving additive maps on B(H)

讨论了B(A)上保交换零积的可加映射,其中A是Banach空间X上的标准算子代数。假设Ф是一个保单位的可加满射。若Ф保交换零积并且对所有的一秩幂等算子P∈A都有Ф(FP)FФ(P)和成立。则有下列四种形式之一:代数同构,共轭代数同构,代数反同构,共轭代数反同构。

It is considered that additive commutative zero-product preserving maps on the algebraA,B of standard operator algebras,acting on real or complex Babach spaces H.Assume that Ф:A→B is an unital additive surjective map.If Ф preserves commutative zero-products and Ф(FP)■FФ(P) and Ф(P)≠0 for every rank one idempotent operator P∈A,Then it has one of the following forms:an algebraic isomorphism,a conjugate algebraic isomor-phism,an algebraic anti-isomorphism and a conjugate algebraic anti-isomorphism.