Jordan rigidity of full matrix algebras

Abstract

Let $\mathbb{F}$ be a field of characteristic different from $2$, and let $M_n(\mathbb{F})^+$ denote the Jordan algebra of all $n\times n$ matrices over $\mathbb{F}$ with product $X\circ Y:=(XY+YX)/2$. We prove a rigidity theorem for $M_n(\mathbb{F})^+$, $n\ge2$: if $\mathcal{J}$ is any $2$-torsion-free Jordan ring and $\phi:M_n(\mathbb{F})^+\to\mathcal{J}$ is a Jordan multiplicative (product-preserving) map, then $\phi(0)$ is an idempotent and $X\mapsto\phi(X)-\phi(0)$ is either zero or an injective Jordan ring homomorphism. Thus, up to an idempotent constant, preservation of the Jordan product alone forces additivity and the zero-or-injective dichotomy. When specialized to associative codomains, the theorem yields the Jacobson--Rickart decomposition into homomorphic and antihomomorphic parts. In particular, for maps $M_n(\mathbb F)^+\to M_k(\mathbb K)^+$, where $\mathbb K$ is a field of characteristic different from $2$, we also obtain a block normal form governed by finite-dimensional $\mathbb K$-representations of $\mathbb F$, together with a criterion for the existence of nonconstant maps.