II₁型因子上关于酉元正交基的Kadison问题的不可分情形
The solution to Kadison's problem on orthonormal bases of unitaries for type $\mathrm{II}_1$ factors
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中文总结 AI 辅助
本文解决了Kadison关于II₁型因子酉元正交基的问题的不可分情形，结合多种引理与方案完成证明，完整解决了该问题。
中文摘要 AI 辅助
1967年，Kadison提出问题：“每个II₁型因子是否都存在一个关于迹的正交基，该基由酉元构成？”在之前的论文[HTZ26]中，He、Tang和Zhang解决了该问题的可分情形。本文证明了互补的不可分情形，从而完整解决了Kadison问题。事实上，该基可选取自伴酉元构成。证明结合了小密度约束下的相对范数引理、确保相关希尔伯特空间投影由环境因子的有界元表示的有限层认证方案，以及沿L²(M,τ)的密度特征的超限扩张。
英文摘要
In 1967, Kadison asked whether every type $\mathrm{II}_1$ factor admits an orthonormal basis, with respect to its trace, consisting of unitaries. We resolve this problem in full generality and, more broadly, characterize the diffuse finite von Neumann algebras admitting such bases consisting of symmetries. Let $M$ be a diffuse finite von Neumann algebra with a faithful normal tracial state $τ$, let $κ$ be the density character of $L^2(M,τ)$, and identify $M$ with its canonical image in $L^2(M,τ)$. We prove that there exists a family $B\subset {s\in M:s=s^*=s^{-1},\ τ(s)=0}$ such that ${1}\cup B$ is an orthonormal basis of $L^2(M,τ)$ if and only if the density character of $L^2(zM,τ(z)^{-1}τ|_{zM})$ equals $κ$ for every nonzero central projection $z\in Z(M)$. In particular, every type $\mathrm{II}_1$ factor admits an orthonormal basis consisting of unitaries, thereby answering Kadison's question affirmatively. We also provide a Lean 4 formalization of the main results.
发表机构
- Fudan University（复旦大学）
- University of Science and Technology of China（中国科学技术大学）
- Xi’an Jiaotong University（西安交通大学）
机构由 AI 辅助整理，请以论文原文为准。
