Intermediate Subalgebras of Cartan embeddings in rings and C*-algebras
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917847699030016 |
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| author | Brown, Jonathan H. Clark, Lisa Orloff Fuller, Adam H. |
| author_facet | Brown, Jonathan H. Clark, Lisa Orloff Fuller, Adam H. |
| contents | Let $D \subseteq A$ be a quasi-Cartan pair of algebras. Then there exists a unique discrete groupoid twist $Σ\to G$ whose twisted Steinberg algebra is isomorphic to $A$ in a way that preserves $D$. In this paper, we show there is a lattice isomorphism between wide open subgroupoids of $G$ and subalgebras $C$ such that $D\subseteq C\subseteq A$ and $D \subseteq C$ is a quasi-Cartan pair. We also characterise which algebraic diagonal/algebraic Cartan/quasi-Cartan pairs have the property that every subalgebra $C$ with $D\subseteq C\subseteq A$ has $D \subseteq C$ a diagonal/Cartan/quasi-Cartan pair. In the diagonal case, when the coefficient ring is a field, it is all of them. Beyond that, only pairs that are close to being diagonal have this property. We then apply our techniques to C*-algebraic inclusions and give a complete characterization of which Cartan pairs $D \subseteq A$ have the property that every C*-subalgebra $C$ with $D\subseteq C\subseteq A$ has $D \subseteq C$ a Cartan pair. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_15924 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Intermediate Subalgebras of Cartan embeddings in rings and C*-algebras Brown, Jonathan H. Clark, Lisa Orloff Fuller, Adam H. Rings and Algebras Operator Algebras Let $D \subseteq A$ be a quasi-Cartan pair of algebras. Then there exists a unique discrete groupoid twist $Σ\to G$ whose twisted Steinberg algebra is isomorphic to $A$ in a way that preserves $D$. In this paper, we show there is a lattice isomorphism between wide open subgroupoids of $G$ and subalgebras $C$ such that $D\subseteq C\subseteq A$ and $D \subseteq C$ is a quasi-Cartan pair. We also characterise which algebraic diagonal/algebraic Cartan/quasi-Cartan pairs have the property that every subalgebra $C$ with $D\subseteq C\subseteq A$ has $D \subseteq C$ a diagonal/Cartan/quasi-Cartan pair. In the diagonal case, when the coefficient ring is a field, it is all of them. Beyond that, only pairs that are close to being diagonal have this property. We then apply our techniques to C*-algebraic inclusions and give a complete characterization of which Cartan pairs $D \subseteq A$ have the property that every C*-subalgebra $C$ with $D\subseteq C\subseteq A$ has $D \subseteq C$ a Cartan pair. |
| title | Intermediate Subalgebras of Cartan embeddings in rings and C*-algebras |
| topic | Rings and Algebras Operator Algebras |
| url | https://arxiv.org/abs/2411.15924 |