Jordan embeddings and linear rank preservers of structural matrix algebras
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929597625401344 |
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| author | Gogić, Ilja Tomašević, Mateo |
| author_facet | Gogić, Ilja Tomašević, Mateo |
| contents | We consider subalgebras $\mathcal{A}$ of the algebra $M_n$ of $n \times n$ complex matrices that contain all diagonal matrices, known in the literature as the structural matrix algebras (SMAs). Let $\mathcal{A} \subseteq M_n$ be an arbitrary SMA. We first show that any commuting family of diagonalizable matrices in $\mathcal{A}$ can be intrinsically simultaneously diagonalized (i.e. the corresponding similarity can be chosen from $\mathcal{A}$). Using this, we then characterize when one SMA Jordan-embeds into another and in that case we describe the form of such Jordan embeddings. As a consequence, we obtain a description of Jordan automorphisms of SMAs, generalizing Coelho's result on their algebra automorphisms. Next, motivated by the results of Marcus-Moyls and Molnar-Šemrl, connecting the linear rank-one preservers with Jordan embeddings $M_n \to M_n$ and $\mathcal{T}_n \to M_n$ (where $\mathcal{T}_n$ is the algebra of $n \times n$ upper-triangular matrices) respectively, we show that any linear unital rank-one preserver $\mathcal{A} \to M_n$ is necessarily a Jordan embedding. As the converse fails in general, we also provide a necessary and sufficient condition for when it does hold true. Finally, we obtain a complete description of linear rank preservers $\mathcal{A} \to M_n$, as maps of the form $X\mapsto S\left(PX + (I-P)X^t\right)T$, for some invertible matrices $S,T \in M_n$ and a central idempotent $P\in\mathcal{A}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_16906 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Jordan embeddings and linear rank preservers of structural matrix algebras Gogić, Ilja Tomašević, Mateo Rings and Algebras 47B49, 16S50, 16W20, 15A20 We consider subalgebras $\mathcal{A}$ of the algebra $M_n$ of $n \times n$ complex matrices that contain all diagonal matrices, known in the literature as the structural matrix algebras (SMAs). Let $\mathcal{A} \subseteq M_n$ be an arbitrary SMA. We first show that any commuting family of diagonalizable matrices in $\mathcal{A}$ can be intrinsically simultaneously diagonalized (i.e. the corresponding similarity can be chosen from $\mathcal{A}$). Using this, we then characterize when one SMA Jordan-embeds into another and in that case we describe the form of such Jordan embeddings. As a consequence, we obtain a description of Jordan automorphisms of SMAs, generalizing Coelho's result on their algebra automorphisms. Next, motivated by the results of Marcus-Moyls and Molnar-Šemrl, connecting the linear rank-one preservers with Jordan embeddings $M_n \to M_n$ and $\mathcal{T}_n \to M_n$ (where $\mathcal{T}_n$ is the algebra of $n \times n$ upper-triangular matrices) respectively, we show that any linear unital rank-one preserver $\mathcal{A} \to M_n$ is necessarily a Jordan embedding. As the converse fails in general, we also provide a necessary and sufficient condition for when it does hold true. Finally, we obtain a complete description of linear rank preservers $\mathcal{A} \to M_n$, as maps of the form $X\mapsto S\left(PX + (I-P)X^t\right)T$, for some invertible matrices $S,T \in M_n$ and a central idempotent $P\in\mathcal{A}$. |
| title | Jordan embeddings and linear rank preservers of structural matrix algebras |
| topic | Rings and Algebras 47B49, 16S50, 16W20, 15A20 |
| url | https://arxiv.org/abs/2409.16906 |