A local quantization principle for inclusions of tracial von Neumann algebras
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866913930382671872 |
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| author | Cao, Xinyan Fang, Junsheng Jiang, Chunlan Yao, Zhaolin |
| author_facet | Cao, Xinyan Fang, Junsheng Jiang, Chunlan Yao, Zhaolin |
| contents | We study the local quantization principle (after Sorin Popa~\cite{popa 94} and \cite{popa 95}) of inclusions of tracial von Neumann algebras. Let $(\mathcal{M},τ)$ be a type ${\rm II}_1$ von Neumann algebra and let $\mathcal{N}\subseteq \mathcal{M}$ be a type ${\rm II}_1$ von Neumann subalgebra. Let $x_1,\ldots, x_m \in \mathcal{M}$ and $ ε> 0$. Then there exists a partition of 1 with projections $p_{1}, \ldots, p_{n}$ in $\mathcal{N}$ such that \[\left\|\sum_{i=1}^n p_{i}\left(x_j-E_{\mathcal{N}'\cap \mathcal{M}}(x_j)\right)p_{i}\right\|_{2}<ε,\quad 1\leq j\leq m.\] In particular, if $\mathcal{N}\subseteq \mathcal{M}$ is an inclusion of type $\rm II_{1}$ factors with $[\mathcal{M}:\mathcal{N}]=2$, then for any $x_{1},\ldots, x_{m}\in \mathcal{M}$, there exists a partition of 1 with projections $p_{1}, \ldots, p_{n}$ in $\mathcal{N}$ such that \[\sum_{i=1}^n p_ix_jp_i=τ(x_j)1, \quad 1\leq j\leq m.\] Equivalently, there exists a unitary operator $u\in \mathcal{N}$ such that \[\frac{1}{n}\sum_{i=1}^nu^{*i}x_j u^i=τ(x_j)1, \quad 1\leq j\leq m.\] |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_04244 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A local quantization principle for inclusions of tracial von Neumann algebras Cao, Xinyan Fang, Junsheng Jiang, Chunlan Yao, Zhaolin Operator Algebras We study the local quantization principle (after Sorin Popa~\cite{popa 94} and \cite{popa 95}) of inclusions of tracial von Neumann algebras. Let $(\mathcal{M},τ)$ be a type ${\rm II}_1$ von Neumann algebra and let $\mathcal{N}\subseteq \mathcal{M}$ be a type ${\rm II}_1$ von Neumann subalgebra. Let $x_1,\ldots, x_m \in \mathcal{M}$ and $ ε> 0$. Then there exists a partition of 1 with projections $p_{1}, \ldots, p_{n}$ in $\mathcal{N}$ such that \[\left\|\sum_{i=1}^n p_{i}\left(x_j-E_{\mathcal{N}'\cap \mathcal{M}}(x_j)\right)p_{i}\right\|_{2}<ε,\quad 1\leq j\leq m.\] In particular, if $\mathcal{N}\subseteq \mathcal{M}$ is an inclusion of type $\rm II_{1}$ factors with $[\mathcal{M}:\mathcal{N}]=2$, then for any $x_{1},\ldots, x_{m}\in \mathcal{M}$, there exists a partition of 1 with projections $p_{1}, \ldots, p_{n}$ in $\mathcal{N}$ such that \[\sum_{i=1}^n p_ix_jp_i=τ(x_j)1, \quad 1\leq j\leq m.\] Equivalently, there exists a unitary operator $u\in \mathcal{N}$ such that \[\frac{1}{n}\sum_{i=1}^nu^{*i}x_j u^i=τ(x_j)1, \quad 1\leq j\leq m.\] |
| title | A local quantization principle for inclusions of tracial von Neumann algebras |
| topic | Operator Algebras |
| url | https://arxiv.org/abs/2507.04244 |