Polynomial preserving maps on certain Jordan algebras

Abstract

LetB andQ be associative algebras and letS be a Jordan subalgebra ofB. Letf(x 1,…,x m ) be a (noncommutative) multilinear polynomial such thatS is closed underf. Letα:S→Q be anf-homomorphism in the sense that it is a linear map preservingf. Under suitable conditions it is shown thatα is essentially given by a ring homomorphism. An analogous theorem forf-derivations is also proved. The proofs rest heavily on results concerning functional identities andd-freeness.