An extension of Petek-Šemrl preserver theorems for Jordan embeddings of structural matrix algebras
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866913743512797184 |
|---|---|
| author | Gogić, Ilja Tomašević, Mateo |
| author_facet | Gogić, Ilja Tomašević, Mateo |
| contents | Let $M_n$ be the algebra of $n \times n$ complex matrices and $\mathcal{T}_n \subseteq M_n$ the corresponding upper-triangular subalgebra. In their influential work, Petek and Šemrl characterize Jordan automorphisms of $M_n$ and $\mathcal{T}_n$, when $n \geq 3$, as (injective in the case of $\mathcal{T}_n$) continuous commutativity and spectrum preserving maps $ϕ: M_n \to M_n$ and $ϕ: \mathcal{T}_n \to \mathcal{T}_n$. Recently, in a joint work with Petek, the authors extended this characterization to the maps $ϕ: \mathcal{A} \to M_n$, where $\mathcal{A}$ is an arbitrary subalgebra of $M_n$ that contains $\mathcal{T}_n$. In particular, any such map $ϕ$ is a Jordan embedding and hence of the form $ϕ(X)=TXT^{-1}$ or $ϕ(X)=TX^tT^{-1}$, for some invertible matrix $T\in M_n$. In this paper we further extend the aforementioned results in the context of structural matrix algebras (SMAs), i.e. subalgebras $\mathcal{A}$ of $M_n$ that contain all diagonal matrices. More precisely, we provide both a necessary and sufficient condition for an SMA $\mathcal{A}\subseteq M_n$ such that any injective continuous commutativity and spectrum preserving map $ϕ: \mathcal{A} \to M_n$ is necessarily a Jordan embedding. In contrast to the previous cases, such maps $ϕ$ no longer need to be multiplicative/antimultiplicative, nor rank-one preservers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_11092 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An extension of Petek-Šemrl preserver theorems for Jordan embeddings of structural matrix algebras Gogić, Ilja Tomašević, Mateo Rings and Algebras 47B49, 15A27, 16S50, 16W20 Let $M_n$ be the algebra of $n \times n$ complex matrices and $\mathcal{T}_n \subseteq M_n$ the corresponding upper-triangular subalgebra. In their influential work, Petek and Šemrl characterize Jordan automorphisms of $M_n$ and $\mathcal{T}_n$, when $n \geq 3$, as (injective in the case of $\mathcal{T}_n$) continuous commutativity and spectrum preserving maps $ϕ: M_n \to M_n$ and $ϕ: \mathcal{T}_n \to \mathcal{T}_n$. Recently, in a joint work with Petek, the authors extended this characterization to the maps $ϕ: \mathcal{A} \to M_n$, where $\mathcal{A}$ is an arbitrary subalgebra of $M_n$ that contains $\mathcal{T}_n$. In particular, any such map $ϕ$ is a Jordan embedding and hence of the form $ϕ(X)=TXT^{-1}$ or $ϕ(X)=TX^tT^{-1}$, for some invertible matrix $T\in M_n$. In this paper we further extend the aforementioned results in the context of structural matrix algebras (SMAs), i.e. subalgebras $\mathcal{A}$ of $M_n$ that contain all diagonal matrices. More precisely, we provide both a necessary and sufficient condition for an SMA $\mathcal{A}\subseteq M_n$ such that any injective continuous commutativity and spectrum preserving map $ϕ: \mathcal{A} \to M_n$ is necessarily a Jordan embedding. In contrast to the previous cases, such maps $ϕ$ no longer need to be multiplicative/antimultiplicative, nor rank-one preservers. |
| title | An extension of Petek-Šemrl preserver theorems for Jordan embeddings of structural matrix algebras |
| topic | Rings and Algebras 47B49, 15A27, 16S50, 16W20 |
| url | https://arxiv.org/abs/2411.11092 |