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Classification of Jordan multiplicative maps on matrix algebras

Classification of Jordan multiplicative maps on matrix algebras

来源:Arxiv_logoArxiv
英文摘要

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 and equal to a fixed idempotent, 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 $2$.

Ilja Gogić、Mateo Tomašević

10.1007/s00010-025-01208-y

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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