Jordan homomorphisms and T-ideals

TL;DR

This paper investigates the structure of Jordan homomorphisms between associative algebras, establishing conditions under which they decompose into homomorphisms and antihomomorphisms, and explores related T-ideals and involutions.

Contribution

It proves that under certain conditions, Jordan epimorphisms decompose into homomorphisms and antihomomorphisms, and characterizes Jordan homomorphisms from involution subalgebras via T-ideals.

Findings

Jordan epimorphisms decompose into homomorphisms and antihomomorphisms when the algebra is unital and equals its commutator ideal.

Existence of a T-ideal allowing extension of Jordan homomorphisms to associative homomorphisms.

Counter-example shows the necessity of the trivial annihilator condition.

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 $\varphi:A\to B$ is the sum of a homomorphism and an antihomomorphism. Our second main result concerns (not necessarily surjective) Jordan homomorphisms from $H(A,*)$ to $B$, where $*$ is an involution on $A$ and $H(A,*)=\{a\in A\,|\, a^*=a\}$. We show that there exists a ${\rm T}$-ideal $G$ having the following two properties: (1) the Jordan homomorphism $\varphi:H(G(A),*)\to B$ can be extended to an (associative) homomorphism, subject to the condition that the subalgebra generated by $\varphi(H(A,*))$ has trivial annihilator, and (2) every element of the ${\rm T}$-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.