关于子因子的铺设尺寸

Pub Date : 2024-01-30 DOI:10.4310/pamq.2023.v19.n5.a6
Sora Popin
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引用次数: 0

摘要

给定一个包含 $\mathrm{II}_1$ 因子 $N \subset M$ 的有限琼斯指数,$[M:N] \lt \infty$,我们证明对于任意 $F \subset M$ 有限且 $\varepsilon \gt 0$,存在一个 $r \leq \lceil 16 \varepsilon^{-2} \rceil \cdot {lceil 4 [M:投影 $p_1, dotsc , p_r 在 N$ 中,这样 ${\lVert \sum^r_{i=1} p_i xp_i - E_{N^\prime \cap M} (x) \rVert}\leq \varepsilon {\lVert x - E_{N^\prime \cap M} (x) \rVert}, \forall x \in F$ (其中 $\lceil \beta \rceil$ 表示最小整数 $\geq \beta$)。我们考虑 $N \subset M$ 的一系列相关不变式,一般称为铺垫大小。
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On the paving size of a subfactor
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.
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