On Maps Preserving Zero Jordan Products

Abstract

Abstract. Let R be a ring, A ¼ M n ðRÞ and : A ! A a surjective additive map preserving zero Jordan products, i.e. if x; y 2 A are such that xy þ yx ¼ 0, then ðxÞ ðyÞ þ ðyÞ ðxÞ ¼ 0. In this paper, we show that if R contains 1 2 and n 5 4, then ¼ ', where ¼ ð1Þ is a central element of A and ' : A ! A is a Jordan homomorphism.2000 Mathematics Subject Classification: 15A04, 47B49

Zero product determined Jordan algebras, I

“…The question whether the full matrix algebra equipped with the Jordan product is zero product determined was examined already in [4]; in the proof we have relied heavily on the method from [6]. In this article and in the forthcoming article we will examine the other most natural examples of Jordan algebras.…”

On maps preserving square-zero matrices

“…In a forthcoming paper [17], we show that a surjective additive map preserving zero Jordan product of matrices over any unital ring must be a scalar multiple of a Jordan homomorphism.…”

Fixed product preserving mappings on Banach algebras

“…Some relevant problems for the Lie product can be found in [18,21,26,28]. For the Jordan product, see [8,13]; most notably, the authors in [8] obtained a complete description for maps preserving equal fixed Jordan products on M n (C). However, the Lie case can be pathological (such as in [18]).…”