Nijenhuis operators on homogeneous spaces related to $C^*$-algebras
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909564427829248 |
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| author | Goliński, Tomasz Larotonda, Gabriel Tumpach, Alice Barbora |
| author_facet | Goliński, Tomasz Larotonda, Gabriel Tumpach, Alice Barbora |
| contents | For a unital non-simple $C^*$-algebra $\mathcal A$ we consider its Banach--Lie group $G$ of invertible elements. For a given closed ideal $\mathfrak k$ in $\mathcal A$, we consider the embedded Banach--Lie subgroup $K$ of $G$ of elements differing from the unit element by an element in $\mathfrak k$. We study vector bundle maps of the tangent space of the homogeneous space $G/K$, induced by an admissible bounded operator on $\mathcal A$. In particular, we discuss when this vector bundle map is a Nijenhuis operator in $G/K$. The special case of almost complex structures in $G/K$ is also addressed. Examples for particular classes of $C^*$-algebras are presented, including the Toeplitz algebra and crossed products by $\mathbb Z$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_22055 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Nijenhuis operators on homogeneous spaces related to $C^*$-algebras Goliński, Tomasz Larotonda, Gabriel Tumpach, Alice Barbora Differential Geometry Functional Analysis Operator Algebras 58B20, 53C30, 53C15, 46L99, 47L80, 47L65 For a unital non-simple $C^*$-algebra $\mathcal A$ we consider its Banach--Lie group $G$ of invertible elements. For a given closed ideal $\mathfrak k$ in $\mathcal A$, we consider the embedded Banach--Lie subgroup $K$ of $G$ of elements differing from the unit element by an element in $\mathfrak k$. We study vector bundle maps of the tangent space of the homogeneous space $G/K$, induced by an admissible bounded operator on $\mathcal A$. In particular, we discuss when this vector bundle map is a Nijenhuis operator in $G/K$. The special case of almost complex structures in $G/K$ is also addressed. Examples for particular classes of $C^*$-algebras are presented, including the Toeplitz algebra and crossed products by $\mathbb Z$. |
| title | Nijenhuis operators on homogeneous spaces related to $C^*$-algebras |
| topic | Differential Geometry Functional Analysis Operator Algebras 58B20, 53C30, 53C15, 46L99, 47L80, 47L65 |
| url | https://arxiv.org/abs/2410.22055 |