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 $\varphi :M_n(\mathbb{F})\to M_n(\mathbb{F})$ is Jordan multiplicative, i.e.\ it satisfies the functional equation $$\varphi (XY+YX)=\varphi (X)\varphi (Y)+\varphi (Y)\varphi (X),\text{for all }X,Y\in M_n(\mathbb{F})$$ if and only if one of the following holds: either $\varphi$ 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 $$\varphi (X)=T\omega (X)T^{-1}\text{ or }\varphi (X)=T\omega (X{)}^t{T}^{-1},\text{for all }X\in M_n(\mathbb{F}),$$ where $\omega (X)$ denotes the matrix obtained by applying $\omega$ entrywise to $X$. In particular, any Jordan multiplicative map $\varphi :M_n(\mathbb{F})\to M_n(\mathbb{F})$ with $\varphi (0)=0$ is automatically additive. The analogous characterization fails when $\mathbb{F}$ has characteristic $2$.

arXiv comment14 pages, closely related to [arxiv.org/abs/2503.14116], to appear in Aequationes Math. In v3 we corrected a minor issue in the abstract and in the statement of Theorem 1.1: the constant term in the characterization should be $P/2$ for $\diamond$-preserving maps, and $P$ for $\circ$-preserving maps, where $P\in M_n(\mathbb{F})$ is an idempotent

References in corpus (1)

• Jordan embeddings and linear rank preservers of structural matrix algebras