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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.08072 |
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| _version_ | 1866918131599933440 |
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| author | Gao, David Elayavalli, Srivatsav Kunnawalkam Patchell, Gregory Tan, Hui |
| author_facet | Gao, David Elayavalli, Srivatsav Kunnawalkam Patchell, Gregory Tan, Hui |
| contents | We study conjugacy orbits of certain types of subalgebras in tracial von Neumann algebras. For any separable II$_1$ factor $N_0$ we construct a highly indecomposable non Gamma II$_1$ factor $N$ such that $N_0 \subset N$ and moreover every von Neumann subalgebra of $N$ with Haagerup's property admits a unique embedding up to unitary conjugation. Such a factor necessarily has to be non separable, but we show that it can be taken of density character $2^{\aleph_0}$. On the other hand we are able to construct for any separable II$_1$ factor $M_0$, a separable II$_1$ factor $M$ containing $M_0$ such that every property (T) subfactor admits a unique embedding into $M$ up to uniformly approximate unitary equivalence; i.e., any pair of embeddings can be conjugated up to a small uniform $2$-norm perturbation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_08072 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On conjugacy and perturbation of subalgebras Gao, David Elayavalli, Srivatsav Kunnawalkam Patchell, Gregory Tan, Hui Operator Algebras Group Theory Logic We study conjugacy orbits of certain types of subalgebras in tracial von Neumann algebras. For any separable II$_1$ factor $N_0$ we construct a highly indecomposable non Gamma II$_1$ factor $N$ such that $N_0 \subset N$ and moreover every von Neumann subalgebra of $N$ with Haagerup's property admits a unique embedding up to unitary conjugation. Such a factor necessarily has to be non separable, but we show that it can be taken of density character $2^{\aleph_0}$. On the other hand we are able to construct for any separable II$_1$ factor $M_0$, a separable II$_1$ factor $M$ containing $M_0$ such that every property (T) subfactor admits a unique embedding into $M$ up to uniformly approximate unitary equivalence; i.e., any pair of embeddings can be conjugated up to a small uniform $2$-norm perturbation. |
| title | On conjugacy and perturbation of subalgebras |
| topic | Operator Algebras Group Theory Logic |
| url | https://arxiv.org/abs/2403.08072 |