发表机构
Indian Institute of Technology Kanpur（坎普尔印度理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究 von Neumann 子代数间的 Mashood-Taylor 与 Kadison-Kastler 距离，证明自旋模型子因子距离公式、交换平方与最大距离的等价性，并在自由群因子中构造最大距离的 masas。
AI 中文摘要
我们通过 Mashood-Taylor（$\mathrm{d}_{\mathrm{MT}}$）距离、Kadison-Kastler（$\mathrm{d}_{\mathrm{KK}}$）距离及其内角来研究 von Neumann 子代数的相对位置。对于超有限 $\mathrm{II}_1$-因子 $\mathscr{R}$ 的 $2\times2$ 自旋模型子因子连续族 $(\mathscr{R}_{\mathsf{H}_\alpha})_{\alpha\in[0,\pi)}$，我们建立了公式 $\mathrm{d}_{\mathrm{MT}}(\mathscr{R}_{\mathsf{H}_\alpha},\mathscr{R}_{\mathsf{H}_\beta}) =|\sin(\alpha-\beta)|$。我们证明，两个这样的子因子在其交集上构成交换平方当且仅当它们距离最大（$\mathrm{d}_{\mathrm{MT}}=1$）。更一般地，我们证明在自然指标条件下，$\mathrm{II}_1$-因子的交换平方迫使距离最大，从而得出 $\mathrm{d}_{\mathrm{KK}}=1=\mathrm{d}_{\mathrm{MT}}$。我们还证明了在 Popa 意义下正交的弥散子代数是最大距离的。相反，$(\mathscr{R}_{\mathsf{H}_\alpha})_{\alpha\in[0,\pi)}$ 中的任意两个成员都不是 Popa 正交的。然而，当两个成员距离最大时，它们在其交集上的内角为 $\pi/2$，这表明通过内角定义的正交性与 Popa 正交性不同。最后，在自由群因子 $L(\mathbb{F}_2)=L(\langle a,b\rangle)$ 中，对于 $u\in L(\langle b\rangle)$，我们建立了公式 $\mathrm{d}_{\mathrm{MT}}(L(\langle a\rangle),uL(\langle a\rangle)u^*) =\sqrt{1-|\tau(u)|^4}$。最后，我们在 $L(\mathbb{F}_2)$ 中构造了不源于 $\mathbb{F}_2$ 子群的最大距离 masas。
英文摘要
We investigate the relative position of von Neumann subalgebras through the Mashood--Taylor ($\mathrm{d}_{\mathrm{MT}}$) and Kadison--Kastler ($\mathrm{d}_{\mathrm{KK}}$) distances and the interior angle. For each $n\in\mathbb{N}$, we show that the hyperfinite $\mathrm{II}_1$-factor $\mathscr{R}$ contains an uncountable family of pairwise distinct regular $n\times n$ spin model subfactors. In particular, this yields a continuous family $(\mathscr{R}_{\mathsf{H}_α})_{α\in[0,π)}$ of $2\times2$ spin model subfactors which, equipped with $\mathrm{d}_{\mathrm{MT}}$, is homeomorphic to the circle group $\mathbb{T}$ and admits a natural topological group structure. We further establish \[ \mathrm{d}_{\mathrm{MT}}(\mathscr{R}_{\mathsf{H}_α},\mathscr{R}_{\mathsf{H}_β}) =\vert\sin(α-β)\vert. \] We prove that two such spin model subfactors form a commuting square over their intersection if and only if they are maximally distant. More generally, commuting squares of $\mathrm{II}_1$-factors, under natural index conditions, force maximal distance, yielding $\mathrm{d}_{\mathrm{KK}}=1=\mathrm{d}_{\mathrm{MT}}$. We also show that diffuse subalgebras which are orthogonal in the sense of Popa are maximally distant. In contrast, no two members of the above spin model family are Popa-orthogonal. Nevertheless, maximal distance between two such subfactors implies that their interior angle over the intersection is $π/2$, highlighting the distinction between angle orthogonality and Popa orthogonality. Finally, in $L(\mathbb{F}_2)=L(\langle a,b\rangle)$, for $u\in L(\langle b\rangle)$, we obtain \[ \mathrm{d}_{\mathrm{MT}}(L(\langle a\rangle),uL(\langle a\rangle)u^*) =\sqrt{1-\vertτ(u)\vert^4}, \] and construct maximally distant masas not arising from subgroups of $\mathbb{F}_2$.
Comments34 pages; A group structure on the space of spin model subfactors added
