Classification of Jordan multiplicative maps on matrix algebras

Abstract

Let $M_n(\mathbb{F})$ be the algebra of $n \times n$ matrices over a field $\mathbb{F}$ of characteristic not equal to $2$. If $n\ge 2$, we show that an arbitrary map $\phi : M_n(\mathbb{F}) \to M_n(\mathbb{F})$ is Jordan multiplicative, i.e.\ it satisfies the functional equation $$ \phi(XY+YX)=\phi(X)\phi(Y)+\phi(Y)\phi(X), \quad \text{for all } X,Y \in M_n(\mathbb{F}) $$ if and only if one of the following holds: either $\phi$ is constant, equal to $P/2$ for some idempotent $P \in M_n(\mathbb{F})$, or there exists an invertible matrix $T \in M_n(\mathbb{F})$ and a ring monomorphism $\omega: \mathbb{F} \to \mathbb{F}$ such that $$ \phi(X)=T\omega(X)T^{-1} \quad \text{ or } \quad \phi(X)=T\omega(X)^tT^{-1}, \quad \text{for all } X \in M_n(\mathbb{F}), $$ where $\omega(X)$ denotes the matrix obtained by applying $\omega$ entrywise to $X$. In particular, any Jordan multiplicative map $\phi : M_n(\mathbb{F}) \to M_n(\mathbb{F})$ with $\phi(0)=0$ is automatically additive. The analogous characterization fails when $\mathbb{F}$ has characteristic $2$.