AI 中文总结
该研究对有限维C*-代数的正则幺正包含关系给出完全刻画，明确了正则性与酉正则性的条件，推广到一般有限维C*-代数的包含关系并揭示其与正规化子矩阵的关联。
AI 中文摘要
我们对有限维C*-代数的正则（Kumjian和Renault意义下）幺正包含关系给出完全刻画。对于Mₙ(ℂ)的子代数⨁ⱼ(ℂᵈⱼ⊗Iₚⱼ)，我们证明正则性仅依赖于重数pⱼ的相等性，而酉正则性（由第一作者与Silambarasan近期刻画）还要求dⱼ相等；我们通过简化的替代证明得到后者。将此推广到由包含矩阵Λ编码的任意有限维C*-代数的包含关系，我们证明正则性等价于Λ上显式的行/列条件——与第一作者和Silambarasan引入的正规化子矩阵一致，因此那里用于检测酉正则性的工具被证明可一般地刻画正则性；酉正则性则通过进一步的维数相等条件得到。
英文摘要
We give a complete characterization of regular (in the sense of Kumjian and Renault) unital inclusions of finite-dimensional $C^*$-algebras. For subalgebras $\bigoplus_j( \mathbb{M}_{d_j}(\mathbb{C}) \otimes \mathbb{I}_{p_j})$ of $\mathbb{M}_n(\mathbb{C})$, we show that regularity depends only on equality of the multiplicities $p_j$, while unitary regularity---characterized recently by the first author and Silambarasan---additionally requires equality of the $d_j$; we recover the latter via a streamlined alternative proof. Extending this to inclusions of arbitrary finite-dimensional $C^*$-algebras, encoded by an inclusion matrix $Λ$, we show that regularity is equivalent to an explicit row/column condition on $Λ$---coinciding with the normalizer matrix introduced by the first author and Silambarasan ---so that the device used there to detect unitary regularity is shown to characterize regularity in general; unitary regularity is recovered by a further dimension-equality condition.
Comments24 pages
