Jordan homomorphisms and T‐ideals

Abstract Let and be associative algebras over a field with . Our first main result states that if is unital and equal to its commutator ideal, then every Jordan epimorphism is the sum of a homomorphism and an antihomomorphism. Our second main result concerns (not necessarily surjective) Jordan homomorphisms from to , where is an involution on and . We show that there exists a ‐ideal having the following two properties: (1) the Jordan homomorphism can be extended to an (associative) homomorphism, subject to the condition that the subalgebra generated by has trivial annihilator, and (2) every element of the ‐ideal of identities of the algebra of matrices is nilpotent modulo . A similar statement is true for Jordan homomorphisms from to . A counter‐example shows that the assumption on trivial annihilator cannot be removed.