Classification of Jordan multiplicative maps on matrix algebras
Classification of Jordan multiplicative maps on matrix algebras
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 $Ï: M_n(\mathbb{F}) \to M_n(\mathbb{F})$ is Jordan multiplicative, i.e. it satisfies the functional equation $$ Ï(XY+YX)=Ï(X)Ï(Y)+Ï(Y)Ï(X), \quad \text{for all } X,Y \in M_n(\mathbb{F}) $$ if and only if one of the following holds: either $Ï$ is constant and equal to a fixed idempotent, or there exists an invertible matrix $T \in M_n(\mathbb{F})$ and a ring monomorphism $Ï: \mathbb{F} \to \mathbb{F}$ such that $$ Ï(X)=TÏ(X)T^{-1} \quad \text{ or } \quad Ï(X)=TÏ(X)^tT^{-1}, \quad \text{for all } X \in M_n(\mathbb{F}), $$ where $Ï(X)$ denotes the matrix obtained by applying $Ï$ entrywise to $X$. In particular, any Jordan multiplicative map $Ï: M_n(\mathbb{F}) \to M_n(\mathbb{F})$ with $Ï(0)=0$ is automatically additive. The analogous characterization fails when $\mathbb{F}$ has characteristic $2$.
Ilja Gogić、Mateo Tomašević
数学
Ilja Gogić,Mateo Tomašević.Classification of Jordan multiplicative maps on matrix algebras[EB/OL].(2025-08-13)[2025-08-24].https://arxiv.org/abs/2503.24094.点此复制
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