Classification of Jordan multiplicative maps on matrix algebras
Ilja Gogić, Mateo Tomašević
TL;DR
This work classifies Jordan multiplicative self-maps on the full matrix algebra $M_n( )$ for $n\ge 2$ and ${\rm char}( )\neq 2$. The authors show that every such map is either constant (equal to an idempotent, or to half an idempotent in the diamond case) or additive and of the form φ(X)=T ω(X) T^{-1}$ or φ(X)=T ω(X)^{t} T^{-1}$, where $T$ is invertible and $ω:\u001F\to\u001F$ is a ring monomorphism (with $ω$ applied entrywise to matrices). Moreover, if φ(0)=0 then φ is automatically additive. The characteristic-two case invalidates the classification, as nonconstant Jordan multiplicative maps need not be additive there. Overall, the paper provides a complete finite-dimensional characterization, aligning Jordan multiplicative maps with inner automorphisms composed with field monomorphisms (and their transpose), and clarifies automatic additivity in this setting.
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 $φ: M_n(\mathbb{F}) \to M_n(\mathbb{F})$ is Jordan multiplicative, i.e.\ it satisfies the functional equation $$ φ(XY+YX)=φ(X)φ(Y)+φ(Y)φ(X), \quad \text{for all } X,Y \in M_n(\mathbb{F}) $$ if and only if one of the following holds: either $φ$ 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 $ω: \mathbb{F} \to \mathbb{F}$ such that $$ φ(X)=Tω(X)T^{-1} \quad \text{ or } \quad φ(X)=Tω(X)^tT^{-1}, \quad \text{for all } X \in M_n(\mathbb{F}), $$ where $ω(X)$ denotes the matrix obtained by applying $ω$ entrywise to $X$. In particular, any Jordan multiplicative map $φ: M_n(\mathbb{F}) \to M_n(\mathbb{F})$ with $φ(0)=0$ is automatically additive. The analogous characterization fails when $\mathbb{F}$ has characteristic $2$.