AI 中文总结
本文研究Ⅱ₁型因子交换平方有限指标包含的代数不变量与遍历行为，建立正则性等性质的转移规律，刻画酉正规化子，引入相对本征基性质，对比群作用条件下交叉积包含的正则性与奇异性。
AI 中文摘要
本文探讨由Ⅱ₁型因子构成的交换平方形成的有限指标包含的代数不变量与遍历行为。首先，建立正则性、中间子因子格及外尔群在相对交换子与不可约性条件下如何跨交换平方转移。此外，通过一个新的分解定理，当下层包含是正则的时，给出上层包含的酉正规化子的完整刻画。将这些发现应用于离散群诱导的交叉积包含，引入保留正则包含的相对本征基性质。最后，通过证明群作用上的相对弱混合条件迫使交叉积包含为奇异，对此进行对比。
英文摘要
This paper explores the algebraic invariants and ergodic behavior of the finite-index inclusions forming commuting squares of $\text{II}_1$ factors. We begin by establishing how regularity, intermediate-subfactor lattices, and Weyl groups transfer across the commuting square under relative commutant and irreducibility conditions. Furthermore, we provide a complete characterization, via a novel factorization theorem, of the unitary normalizers of the upper inclusion when the lower inclusion is regular. Applying these findings to crossed-product inclusions induced by discrete groups, we introduce a relative eigenbasis property that preserves regular inclusions. Finally, we contrast this by proving that a relative weak mixing condition on the group action forces the crossed-product inclusion to be singular.
Comments17 pages, minor changes
