Orthogonality of Defect Ideals for Jordan Homomorphisms

Abstract

Let $φ\colon A\to B$ be a Jordan homomorphism of associative algebras over a field $\mathbb{F}$ of characteristic different from $2$, with defect ideals $V_φ$ and $W_φ$. If $A=\operatorname{Alg}([[A,A],[A,A]])$, then $V_φW_φ=W_φV_φ=0$ and $V_φ\cap W_φ\subseteq\operatorname{Ann}(\operatorname{Alg}(φ(A)))$. Moreover, the induced map modulo $V_φ\cap W_φ$ is standard. The annihilator bound is sharp, even in finite dimension. As an application, when $A$ is unital and $A=K(A)$, every Jordan homomorphism defined on $A$ is splittable and standard. In this setting, the corresponding result of Brešar and Zelmanov holds without the surjectivity hypothesis.