Orthogonality of Defect Ideals for Jordan Homomorphisms

Let $\phi \text{\hspace{0.17em}⁣}:A\to B$ be a Jordan homomorphism of associative algebras over a field $\mathbb{F}$ of characteristic different from $2$, with defect ideals $V_{\phi }$ and $W_{\phi }$. If $A=\mathrm{Alg}([[A,A],[A,A]])$, then $V_{\phi }{W}_{\phi }=W_{\phi }{V}_{\phi }=0$ and $V_{\phi }\cap W_{\phi }\subseteq \mathrm{Ann}(\mathrm{Alg}(\phi (A)))$. Moreover, the induced map modulo $V_{\phi }\cap W_{\phi }$ 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.