Jordan zero-product preserving additive maps on operator algebras

Abstract

Let Φ : A → B be an additive surjective map between some operator algebras such that A B + B A = 0 implies Φ ( A ) Φ ( B ) + Φ ( B ) Φ ( A ) = 0 . We show that, under some mild conditions, Φ is a Jordan homomorphism multiplied by a central element. Such operator algebras include von Neumann algebras, C ∗ -algebras and standard operator algebras, etc. Particularly, if H and K are infinite-dimensional (real or complex) Hilbert spaces and A = B ( H ) and B = B ( K ) , then there exists a nonzero scalar c and an invertible linear or conjugate-linear operator U : H → K such that either Φ ( A ) = c U A U −1 for all A ∈ B ( H ) , or Φ ( A ) = c U A ∗ U −1 for all A ∈ B ( H ) .