Characterizing Jordan embeddings between block upper-triangular subalgebras via preserving properties
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929550607253504 |
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| author | Gogić, Ilja Petek, Tatjana Tomašević, Mateo |
| author_facet | Gogić, Ilja Petek, Tatjana Tomašević, Mateo |
| contents | Let $M_n$ be the algebra of $n \times n$ complex matrices. We consider arbitrary subalgebras $\mathcal{A}$ of $M_n$ which contain the algebra of all upper-triangular matrices (i.e.\ block upper-triangular subalgebras), and their Jordan embeddings. We first describe Jordan embeddings $ϕ: \mathcal{A} \to M_n$ as maps of the form $ϕ(X)=TXT^{-1}$ or $ϕ(X)=TX^tT^{-1}$, where $T\in M_n$ is an invertible matrix, and then we obtain a simple criteria of when one block upper-triangular subalgebra Jordan-embeds into another (and in that case we describe the form of such embeddings). As a main result, we characterize Jordan embeddings $ϕ: \mathcal{A} \to M_n$ (when $n\geq 3$) as continuous injective maps which preserve commutativity and spectrum. We show by counterexamples that all these assumptions are indispensable (unless $\mathcal{A} = M_n$ when injectivity is superfluous). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_09864 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Characterizing Jordan embeddings between block upper-triangular subalgebras via preserving properties Gogić, Ilja Petek, Tatjana Tomašević, Mateo Rings and Algebras 47B49, 15A27, 16S50, 16W20 Let $M_n$ be the algebra of $n \times n$ complex matrices. We consider arbitrary subalgebras $\mathcal{A}$ of $M_n$ which contain the algebra of all upper-triangular matrices (i.e.\ block upper-triangular subalgebras), and their Jordan embeddings. We first describe Jordan embeddings $ϕ: \mathcal{A} \to M_n$ as maps of the form $ϕ(X)=TXT^{-1}$ or $ϕ(X)=TX^tT^{-1}$, where $T\in M_n$ is an invertible matrix, and then we obtain a simple criteria of when one block upper-triangular subalgebra Jordan-embeds into another (and in that case we describe the form of such embeddings). As a main result, we characterize Jordan embeddings $ϕ: \mathcal{A} \to M_n$ (when $n\geq 3$) as continuous injective maps which preserve commutativity and spectrum. We show by counterexamples that all these assumptions are indispensable (unless $\mathcal{A} = M_n$ when injectivity is superfluous). |
| title | Characterizing Jordan embeddings between block upper-triangular subalgebras via preserving properties |
| topic | Rings and Algebras 47B49, 15A27, 16S50, 16W20 |
| url | https://arxiv.org/abs/2311.09864 |