Rigid Graph Products
Fuente:
arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910209593573376 |
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| author | Borst, Matthijs Caspers, Martijn Chen, Enli |
| author_facet | Borst, Matthijs Caspers, Martijn Chen, Enli |
| contents | We prove rigidity properties for von Neumann algebraic graph products. We introduce the notion of rigid graphs and define a class of II$_1$-factors named $\mathcal{C}_{\rm Rigid}$. For von Neumann algebras in this class we show a unique rigid graph product decomposition. In particular, we obtain unique prime factorization results and unique free product decomposition results for new classes of von Neumann algebras. Furthermore, we show that for many graph products of II$_1$-factors, including the hyperfinite II$_1$-factor, we can, up to a constant 2, retrieve the radius of the graph from the graph product. We also prove several technical results concerning relative amenability and embeddings of (quasi)-normalizers in graph products. Furthermore, we give sufficient conditions for a graph product to be nuclear and characterize strong solidity, primeness and free-indecomposability for graph products. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_06171 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rigid Graph Products Borst, Matthijs Caspers, Martijn Chen, Enli Operator Algebras Functional Analysis We prove rigidity properties for von Neumann algebraic graph products. We introduce the notion of rigid graphs and define a class of II$_1$-factors named $\mathcal{C}_{\rm Rigid}$. For von Neumann algebras in this class we show a unique rigid graph product decomposition. In particular, we obtain unique prime factorization results and unique free product decomposition results for new classes of von Neumann algebras. Furthermore, we show that for many graph products of II$_1$-factors, including the hyperfinite II$_1$-factor, we can, up to a constant 2, retrieve the radius of the graph from the graph product. We also prove several technical results concerning relative amenability and embeddings of (quasi)-normalizers in graph products. Furthermore, we give sufficient conditions for a graph product to be nuclear and characterize strong solidity, primeness and free-indecomposability for graph products. |
| title | Rigid Graph Products |
| topic | Operator Algebras Functional Analysis |
| url | https://arxiv.org/abs/2408.06171 |