发表机构
Indian Institute of Technology Kanpur（印度坎普尔理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究有限维矩阵代数中masas的Kadison-Kastler距离与内角，给出显式公式，并证明最大距离、最大内角、最大修正熵与最小概率指数的等价性，且当相对酉为Hadamard酉时达到极值。
AI 中文摘要
我们研究了有限维矩阵代数中masas之间的Kadison-Kastler距离和内角。在\\(\mathbb{M}_2(\mathbb{C})\\)中，我们得到了两个masas之间Kadison-Kastler距离的显式公式，并证明了\\([0,1]\\)中的每个值都可以由\\(\mathrm{d}_{\textrm{KK}}(\Delta,u\Delta u^*)\\)取得。我们进一步证明了当且仅当相对酉矩阵是Hadamard酉矩阵时，该距离达到其最大值。此外，对于\\(\Delta\\)的群胚正规化子中的每个\\(v\\)，\\(\Delta\\)和\\(v\Delta v^*\\)之间的Kadison-Kastler距离要么取其最小值，要么取其最大值。我们还建立了对于任意两个中间子代数\\(\mathbb{C}\subsetneq\mathcal{A},\mathcal{B}\subsetneq\mathbb{M}_2(\mathbb{C})\\)，恒等式\\(\mathrm{d}_{\textrm{KK}}(\mathcal{A},\mathcal{B})=\sin\alpha(\mathcal{A},\mathcal{B})\\)成立。最值得注意的是，对于任意酉矩阵\\(u,v\in\mathbb{M}_2(\mathbb{C})\\)，我们建立了最大Kadison-Kastler距离、最大内角、最大Connes--Størmer修正熵和最小Popa概率指数的等价性：\\(\mathrm{d}_{\textrm{KK}}(u\Delta u^*,v\Delta v^*)=1 \iff h(u\Delta u^*,v\Delta v^*)=\log 2 \iff \alpha(u\Delta u^*,v\Delta v^*)=\frac{\pi}{2} \iff \lambda(u\Delta u^*,v\Delta v^*)=\frac{1}{2}\\)。对于一般的\\(\mathbb{M}_n(\mathbb{C})\\)，我们证明了内角的酉不变性，并推导了形如\\(u\Delta^{(n)}u^*\\)和\\(v\Delta^{(n)}v^*\\)的任意masas之间角的显式公式。因此，当且仅当相对酉矩阵\\(u^*v\\)是Hadamard酉矩阵时，内角为\\(\frac{\pi}{2}\\)。最后，我们证明了\\(\left[0,\frac{\pi}{2}\right]\\)中的每个值都可以作为\\(\mathbb{M}_n(\mathbb{C})\\)中一对masas之间的内角取得。
英文摘要
We study the Kadison--Kastler distance and the interior angle between masas in finite-dimensional matrix algebras. In \(\mathbb{M}_2(\mathbb{C})\), we obtain an explicit formula for the Kadison--Kastler distance between two masas and show that every value in \([0,1]\) is attained by \(\mathrm{d}_{\textrm{KK}}(Δ,uΔu^*)\). We further prove that the distance attains its maximal value precisely when the relative unitary is a Hadamard unitary. In addition, for every $v$ in the groupoid normaliser of $Δ$, the Kadison--Kastler distance between \(Δ\) and \(vΔv^*\) assumes either its minimal or maximal possible value. We also establish the identity \[ \mathrm{d}_{\textrm{KK}}(\mathcal{A},\mathcal{B})=\sinα(\mathcal{A},\mathcal{B}) \] for any two intermediate subalgebras \(\mathbb{C}\subsetneq\mathcal{A},\mathcal{B}\subsetneq\mathbb{M}_2(\mathbb{C})\). Most notably, for any unitaries \(u,v\in\mathbb{M}_2(\mathbb{C})\), we establish the equivalence of maximal Kadison--Kastler distance, maximal interior angle, maximal Connes--Størmer modified entropy, and minimal Popa probabilistic index: \[ \mathrm{d}_{\textrm{KK}}(uΔu^*,vΔv^*)=1 \iff h(uΔu^*,vΔv^*)=\log 2 \iff α(uΔu^*,vΔv^*)=\fracπ{2} \iff λ(uΔu^*,vΔv^*)=\frac{1}{2}. \] For general \(\mathbb{M}_n(\mathbb{C})\), we prove a unitary invariance property for the interior angle and derive an explicit formula for the angle between arbitrary masas of the form \(uΔ^{(n)}u^*\) and \(vΔ^{(n)}v^*\). As a consequence, the interior angle is \(\fracπ{2}\) precisely when the relative unitary \(u^*v\) is a Hadamard unitary. Finally, we show that every value in \(\left[0,\fracπ{2}\right]\) is attained as the interior angle between a pair of masas in \(\mathbb{M}_n(\mathbb{C})\).
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