子因子正则性
Regularity for Subfactors
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中文总结 AI 辅助
本文研究冯·诺依曼代数包含的正则性概念，证明一般情形下正则性不蕴含酉正则性，但在子因子情形下三者等价，并对简单C*-代数建立了类似等价结果。
中文摘要 AI 辅助
我们研究了含单位元的冯·诺依曼代数包含 $\mathcal{B} \subseteq \mathcal{M}$ 的三个自然相关的正则性概念：正则性、群胚正则性和酉正则性。我们首先观察到，通过证明 $\mathcal{B}$ 在 $\mathcal M$ 中的标准正规化子与群胚正规化子生成相同的线性张成，正则性与群胚正则性无条件等价。本工作研究的核心问题是正则性是否蕴含酉正则性。对于任意的冯·诺依曼子代数，我们证明这一蕴含关系严重失效；我们构造了显式例子，表明每个因子（类型 $\mathrm{I}_\infty$、$\mathrm{II}_1$、$\mathrm{II}_\infty$ 和 $\mathrm{III}$）都包含一个正则但不酉正则的冯·诺依曼子代数。与此形成鲜明对比的是，我们证明在子因子情形下行为完全改变。对于所有这些类型的子因子包含，正则性总是蕴含酉正则性，因此三个正则性概念全部重合。最后，我们为具有有限 Watatani 指标条件期望的简单 $C^*$-代数的不可约含单位元包含建立了类似结果。在此情形下，我们证明正则性、酉正则性、成为广义 Cartan 子代数以及由有限群交叉积产生这四者相互等价。
英文摘要
We study three naturally associated notions of regularity for a unital inclusion of von Neumann algebras $\mathcal{B} \subseteq \mathcal{M}$: regularity, groupoid regularity, and unitary regularity. We first observe that regularity and groupoid regularity are unconditionally equivalent by showing that the standard normaliser and the groupoid normaliser of $\mathcal{B}$ in $\mathcal M$ generate the same linear span. The central question addressed in this work is whether regularity implies unitary regularity. For arbitrary von Neumann subalgebras, we demonstrate that this implication fails dramatically; we construct explicit examples to show that every factor (of type $\mathrm{I}_\infty$, $\mathrm{II}_1$, $\mathrm{II}_\infty$, and $\mathrm{III}$) contains a regular von Neumann subalgebra that is not unitarily regular. In stark contrast, we prove that the behaviour changes completely in the setting of subfactors. For subfactor inclusions across all these types, regularity always implies unitary regularity, and thus all three notions of regularity coincide. Finally, we establish an analogous result for irreducible unital inclusions of simple $C^*$-algebras admitting a conditional expectation of finite Watatani index. In this setting, we prove that regularity, unitary regularity, being a generalised Cartan subalgebra, and arising from a crossed product by a finite group are all mutually equivalent conditions.
发表机构
- Indian Institute of Technology Kanpur（坎普尔印度理工学院）
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