Jordan maps and zero Lie product determined algebras

Authors

MATEJ BRESAR

DOI

10.55730/1300-0098.3226

Abstract

Let $A$ be an algebra over a field $F$ with $(F)\ne 2$. If $A$ is generated as an algebra by $[[A,A],[A,A]]$, then for every skew-symmetric bilinear map $\Phi:A\times A\to X$, where $X$ is an arbitrary vector space over $F$, the condition that $\Phi(x^2,x)=0 $ for all $x\in A$ implies that $\Phi(xy,z) +\Phi(zx,y) + \Phi(yz,x)=0$ for all $x,y,z\in A$. This is applicable to the question of whether $A$ is zero Lie product determined and is also used in proving that a Jordan homomorphism from $A$ onto a semiprime algebra $B$ is the sum of a homomorphism and an antihomomorphism.

Keywords

Bilinear map, zero Lie product determined algebra, derivation, Jordan derivation, Jordan homomorphism, functional identity

First Page

1691

Last Page

1698

Recommended Citation

BRESAR, MATEJ (2022) "Jordan maps and zero Lie product determined algebras," Turkish Journal of Mathematics: Vol. 46: No. 5, Article 5. https://doi.org/10.55730/1300-0098.3226
Available at: https://journals.tubitak.gov.tr/math/vol46/iss5/5