Classification of abelian finite-dimensional $C^*$-algebras by orthogonality
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913570683355136 |
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| author | Kuzma, Bojan Singla, Sushil |
| author_facet | Kuzma, Bojan Singla, Sushil |
| contents | The main goal of the article is to prove that if $\mathcal A_1$ and $\mathcal A_2$ are Birkhoff-James isomorphic $C^*$-algebras over the fields $\mathbb F_1$ and $\mathbb F_2$, respectively and if $\mathcal A_1$ finite-dimensional, abelian of dimension greater than one, then $\mathbb F_1=\mathbb F_2$ and $\mathcal A_1$ and $\mathcal A_2$ are (isometrically) $\ast$-isomorphic $C^*$-algebras. Furthermore, it is also proved that for a finite-dimensional $C^*$-algebra $\mathcal A$, we have $\mathcal L_{\mathcal A}^\bot$ is the sum of minimal ideals which are not skew-fields and $\mathcal L_{\mathcal A}^{\bot\bot}$ is the sum of minimal ideals which are skew-fields, where $\mathcal L_{\mathcal A}$ denotes the set of all left-symmetric elements in $\mathcal A$ and for any subset $\mathcal S\subseteq \mathcal A$, the set $\mathcal S^\bot$ represents the set of all elements of $\mathcal A$ which are Birkhoff-James orthogonal to $\mathcal S$. A procedure to extract the minimal ideals which are (commutative) fields is also given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_01684 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Classification of abelian finite-dimensional $C^*$-algebras by orthogonality Kuzma, Bojan Singla, Sushil Functional Analysis Operator Algebras 46L05, 46B20, 46B80 The main goal of the article is to prove that if $\mathcal A_1$ and $\mathcal A_2$ are Birkhoff-James isomorphic $C^*$-algebras over the fields $\mathbb F_1$ and $\mathbb F_2$, respectively and if $\mathcal A_1$ finite-dimensional, abelian of dimension greater than one, then $\mathbb F_1=\mathbb F_2$ and $\mathcal A_1$ and $\mathcal A_2$ are (isometrically) $\ast$-isomorphic $C^*$-algebras. Furthermore, it is also proved that for a finite-dimensional $C^*$-algebra $\mathcal A$, we have $\mathcal L_{\mathcal A}^\bot$ is the sum of minimal ideals which are not skew-fields and $\mathcal L_{\mathcal A}^{\bot\bot}$ is the sum of minimal ideals which are skew-fields, where $\mathcal L_{\mathcal A}$ denotes the set of all left-symmetric elements in $\mathcal A$ and for any subset $\mathcal S\subseteq \mathcal A$, the set $\mathcal S^\bot$ represents the set of all elements of $\mathcal A$ which are Birkhoff-James orthogonal to $\mathcal S$. A procedure to extract the minimal ideals which are (commutative) fields is also given. |
| title | Classification of abelian finite-dimensional $C^*$-algebras by orthogonality |
| topic | Functional Analysis Operator Algebras 46L05, 46B20, 46B80 |
| url | https://arxiv.org/abs/2411.01684 |