Classification of Jordan multiplicative maps on matrix algebras
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909921867464704 |
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| author | Gogić, Ilja Tomašević, Mateo |
| author_facet | Gogić, Ilja Tomašević, Mateo |
| contents | 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 $2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_24094 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Classification of Jordan multiplicative maps on matrix algebras Gogić, Ilja Tomašević, Mateo Rings and Algebras 47B49, 16S50, 16W20, 20M25 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 $2$. |
| title | Classification of Jordan multiplicative maps on matrix algebras |
| topic | Rings and Algebras 47B49, 16S50, 16W20, 20M25 |
| url | https://arxiv.org/abs/2503.24094 |