{"title":"关于子因子的铺设尺寸","authors":"Sora Popin","doi":"10.4310/pamq.2023.v19.n5.a6","DOIUrl":null,"url":null,"abstract":"Given an inclusion of $\\mathrm{II}_1$ factors $N \\subset M$ with finite Jones index, $[M:N] \\lt \\infty$, we prove that for any $F \\subset M$ finite and $\\varepsilon \\gt 0$, there exists a partition of $1$ with $r \\leq \\lceil 16 \\varepsilon^{-2} \\rceil \\cdot {\\lceil 4 [M:N] \\varepsilon}^{-2} \\rceil$ projections $p_1, \\dotsc , p_r \\in N$ such that ${\\lVert \\sum^r_{i=1} p_i xp_i - E_{N^\\prime \\cap M} (x) \\rVert} \\leq \\varepsilon {\\lVert x - E_{N^\\prime \\cap M} (x) \\rVert}, \\forall x \\in F$ (where $\\lceil \\beta \\rceil$ denotes the least integer $\\geq \\beta$). We consider a series of related invariants for $N \\subset M$, generically called <i>paving size.</i>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the paving size of a subfactor\",\"authors\":\"Sora Popin\",\"doi\":\"10.4310/pamq.2023.v19.n5.a6\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Given an inclusion of $\\\\mathrm{II}_1$ factors $N \\\\subset M$ with finite Jones index, $[M:N] \\\\lt \\\\infty$, we prove that for any $F \\\\subset M$ finite and $\\\\varepsilon \\\\gt 0$, there exists a partition of $1$ with $r \\\\leq \\\\lceil 16 \\\\varepsilon^{-2} \\\\rceil \\\\cdot {\\\\lceil 4 [M:N] \\\\varepsilon}^{-2} \\\\rceil$ projections $p_1, \\\\dotsc , p_r \\\\in N$ such that ${\\\\lVert \\\\sum^r_{i=1} p_i xp_i - E_{N^\\\\prime \\\\cap M} (x) \\\\rVert} \\\\leq \\\\varepsilon {\\\\lVert x - E_{N^\\\\prime \\\\cap M} (x) \\\\rVert}, \\\\forall x \\\\in F$ (where $\\\\lceil \\\\beta \\\\rceil$ denotes the least integer $\\\\geq \\\\beta$). We consider a series of related invariants for $N \\\\subset M$, generically called <i>paving size.</i>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-01-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.4310/pamq.2023.v19.n5.a6\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4310/pamq.2023.v19.n5.a6","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Given an inclusion of $\mathrm{II}_1$ factors $N \subset M$ with finite Jones index, $[M:N] \lt \infty$, we prove that for any $F \subset M$ finite and $\varepsilon \gt 0$, there exists a partition of $1$ with $r \leq \lceil 16 \varepsilon^{-2} \rceil \cdot {\lceil 4 [M:N] \varepsilon}^{-2} \rceil$ projections $p_1, \dotsc , p_r \in N$ such that ${\lVert \sum^r_{i=1} p_i xp_i - E_{N^\prime \cap M} (x) \rVert} \leq \varepsilon {\lVert x - E_{N^\prime \cap M} (x) \rVert}, \forall x \in F$ (where $\lceil \beta \rceil$ denotes the least integer $\geq \beta$). We consider a series of related invariants for $N \subset M$, generically called paving size.