Simplices of maximally amenable extensions in II$_1$ factors
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916440020353024 |
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| author | Elayavalli, Srivatsav Kunnawalkam Patchell, Gregory |
| author_facet | Elayavalli, Srivatsav Kunnawalkam Patchell, Gregory |
| contents | For every $n\in \mathbb{N}$ we obtain a separable II$_1$ factor $M$ and a maximally abelian subalgebra $A\subset M$ such that the space of maximally amenable extensions of $A$ in $M$ is affinely identified with the $n$ dimensional $\mathbb{R}$-simplex. This moreover yields first examples of masas in II$_1$ factors $A\subset M$ admitting exactly $n$ maximally amenable factorial extensions. Our examples of such $M$ are group von Neumann algebras of free products of lamplighter groups amalgamated over the acting group. A conceptual ingredient that goes into obtaining this result is a simultaneous relative asymptotic orthogonality property, extending prior works in the literature. The proof uses technical tools including our uniform-flattening strategy for commutants in ultrapowers of II$_1$ factors. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_11788 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Simplices of maximally amenable extensions in II$_1$ factors Elayavalli, Srivatsav Kunnawalkam Patchell, Gregory Operator Algebras Group Theory For every $n\in \mathbb{N}$ we obtain a separable II$_1$ factor $M$ and a maximally abelian subalgebra $A\subset M$ such that the space of maximally amenable extensions of $A$ in $M$ is affinely identified with the $n$ dimensional $\mathbb{R}$-simplex. This moreover yields first examples of masas in II$_1$ factors $A\subset M$ admitting exactly $n$ maximally amenable factorial extensions. Our examples of such $M$ are group von Neumann algebras of free products of lamplighter groups amalgamated over the acting group. A conceptual ingredient that goes into obtaining this result is a simultaneous relative asymptotic orthogonality property, extending prior works in the literature. The proof uses technical tools including our uniform-flattening strategy for commutants in ultrapowers of II$_1$ factors. |
| title | Simplices of maximally amenable extensions in II$_1$ factors |
| topic | Operator Algebras Group Theory |
| url | https://arxiv.org/abs/2410.11788 |