Multiplicative and Jordan multiplicative maps on structural matrix algebras
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918217697460224 |
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| author | Gogić, Ilja Tomašević, Mateo |
| author_facet | Gogić, Ilja Tomašević, Mateo |
| contents | Let $M_n$ denote the algebra of $n \times n$ complex matrices and let $\mathcal{A}\subseteq M_n$ be an arbitrary structural matrix algebra, i.e. a subalgebra of $M_n$ that contains all diagonal matrices. We consider injective maps $ϕ: \mathcal{A}\to M_n$ that satisfy the condition $$ ϕ(X \bullet Y) = ϕ(X) \bullet ϕ(Y), \quad \text{for all } X,Y \in \mathcal{A}, $$ where $\bullet$ is either the standard matrix multiplication $(X,Y)\mapsto XY$, the Jordan product $(X,Y) \mapsto XY+YX$, or the normalized Jordan product $(X,Y) \mapsto \frac{1}{2}(XY+YX)$. We show that all such maps $ϕ$ are automatically additive if and only if $\mathcal{A}$ does not contain a central rank-one idempotent. Moreover, in this case, we fully characterize the form of these maps. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_14116 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multiplicative and Jordan multiplicative maps on structural matrix algebras Gogić, Ilja Tomašević, Mateo Rings and Algebras 47B49, 16S50, 16W20, 20M25 Let $M_n$ denote the algebra of $n \times n$ complex matrices and let $\mathcal{A}\subseteq M_n$ be an arbitrary structural matrix algebra, i.e. a subalgebra of $M_n$ that contains all diagonal matrices. We consider injective maps $ϕ: \mathcal{A}\to M_n$ that satisfy the condition $$ ϕ(X \bullet Y) = ϕ(X) \bullet ϕ(Y), \quad \text{for all } X,Y \in \mathcal{A}, $$ where $\bullet$ is either the standard matrix multiplication $(X,Y)\mapsto XY$, the Jordan product $(X,Y) \mapsto XY+YX$, or the normalized Jordan product $(X,Y) \mapsto \frac{1}{2}(XY+YX)$. We show that all such maps $ϕ$ are automatically additive if and only if $\mathcal{A}$ does not contain a central rank-one idempotent. Moreover, in this case, we fully characterize the form of these maps. |
| title | Multiplicative and Jordan multiplicative maps on structural matrix algebras |
| topic | Rings and Algebras 47B49, 16S50, 16W20, 20M25 |
| url | https://arxiv.org/abs/2503.14116 |