Zero product determined Jordan algebras, I

Abstract

We show that the Jordan algebra S of symmetric matrices with respect to either transpose or symplectic involution is zero product determined. This means that if a bilinear map {., .} from S Â S into a vector space X satisfies {x, y} ¼ 0 whenever x y ¼ 0, then there exists a linear map T : S ! X such that {x, y} ¼ T(x y) for all x, y 2 S (here, x y ¼ xy þ yx).

Finite dimensional zero product determined algebras are generated by idempotents

“…The result obtained has turned out to be crucial for obtaining a definitive description of commutativity preserving (= zero Lie product preserving) linear maps on finite dimensional central simple algebras. Zero product determined algebras have been accordingly extensively studied in Lie and also Jordan algebras [12,[22][23][24]32]. We do not want to enter the area of general nonassociative algebras in this paper.…”

On Maps Determined by Zero Products

“…In the main result of Section 2 we will actually consider a multiadditive map in an arbitrary number of variables satisfying a condition of this sort (Theorem 2.1). Here we were motivated by several recent works [1,2,7,8,10,11,14] dealing with biadditive maps satisfying some related conditions. An application of Theorem 2.1 to maps satisfying (1) will lead to a functional identity which is not covered by the general theory [6].…”

Zero action determined modules for associative algebras

“…The original motivation for studying zero product determined algebras is to characterise the zero product preserving linear maps, see [8,10,11] for some historic background. After the seminal paper [6], the concept of zero product determined algebra was extensively studied for associative and non-associative algebras (see [9,14,15,16,18] and the reference therein). At the same time, more applications of zero product determined algebras were found [4,18], such as commutativity preserving linear maps, maps derivable at zero, etc.…”