B(H)上保交换零积的可加映射

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

讨论了B(H)上保交换零积的可加映射,其中B(H)是由Hilbert空间H上的有界线性算子全体组成的Banach代数。首先给出了在有限维情形下,若Φ是保交换零积的可加满射,使得Φ(I)=I,并且对每个一秩幂等算子P∈Mn都有Φ(FΦ)FΦ(P),则Φ是一个自同构或反自同构。进一步给出了无限维情形下,若Φ是保交换零积可加满射,则Φ是非零数乘一个环同构或一个环反同构。

It is obtained that additive commutative zero-product preserving maps on the algebra B（H） of all bounded linear operators, acting on the real or the complex Hilbert space H. If H is finite dimensional and φ is a surjective commutative zero-product preserving additive maps on B（H） such that φ{I}=I and φ（Fφ）∪→Fφ（P） for all rank one idempotent operators, then φ is an automorphism or an anti-automorphism. If H is infinite dimensional and φ is a surjective commutative zero-product preserving additive maps on B （H）, It is obtained that φ is a nonzero scalar multiple of a ring isomorphism or a ring anti-isomorphism.