Jordan homomorphisms and T-ideals

Abstract

Let $A$ and $B$ be associative algebras over a field $F$ with {\rm char}$(F)\ne 2$. Our first main result states that if $A$ is unital and equal to its commutator ideal, then every Jordan epimorphism $\ddot{I}†:A\to B$ is the sum of a homomorphism and an antihomomorphism. Our second main result concerns (not necessarily surjective) Jordan homomorphisms from $H(A,\ast )$ to $B$, where $\ast$ is an involution on $A$ and $H(A,\ast )=\{a\in A|a^{\ast }=a\}$. We show that there exists a $\mathrm{T}$-ideal $G$ having the following two properties: (1) the Jordan homomorphism $\ddot{I}†:H(G(A),\ast )\to B$ can be extended to an (associative) homomorphism, subject to the condition that the subalgebra generated by $\ddot{I}†(H(A,\ast ))$ has trivial annihilator, and (2) every element of the $\mathrm{T}$-ideal of identities of the algebra of $2\times 2$ matrices is nilpotent modulo $G$. A similar statement is true for Jordan homomorphisms from $A$ to $B$. A counter-example shows that the assumption on trivial annihilator cannot be removed.