充足C*-对角的对角比较
Diagonal comparison of ample C*-diagonals
AI总结：
本文引入适用于子C*-代数具全不连通谱的对角对的对角比较性质，确立其与环境C*-代数严格比较、底层动力学动态比较的等价性，应用于主极小变换群胚对角对，还证明对角投影一致迹分割等结果。
摘要：
我们引入对角比较这一正则性性质，适用于子C*-代数具有全不连通谱的对角对，并确立其与环境C*-代数的严格比较、正规子群部分作用诱导的底层动力学的动态比较同时成立的等价性。作为应用，我们证明：对于由具有全不连通单位空间的主极小变换群胚产生的对角对，对角比较等价于该对的迹Z稳定性，且有限对角维数可推导出它。在此过程中，我们证明对角子C*-代数的任何投影均可被一致迹分割，并探索了到交换子C*-代数的条件期望的性质，即每个正元包含于其条件期望生成的遗传子代数中。我们还证明：具有有限对角维数的C*-对所关联的期望总是具有该遗传性质，并给出一个不满足该性质的例子。
英文摘要：
We introduce diagonal comparison, a regularity property of diagonal pairs where the sub-C*-algebra has totally disconnected spectrum, and establish its equivalence with the concurrence of strict comparison of the ambient C*-algebra and dynamical comparison of the underlying dynamics induced by the partial action of the normalisers. As an application, we show that for diagonal pairs arising from principal minimal transformation groupoids with totally disconnected unit space, diagonal comparison is equivalent to tracial Z-stability of the pair and that it is implied by finite diagonal dimension. In-between, we show that any projection of the diagonal sub-C*-algebra can be uniformly tracially divided, and explore a property of conditional expectations onto abelian sub-C*-algebras, namely containment of every positive element in the hereditary subalgebra generated by its conditional expectation. We show that the expectation associated to a C*-pair with finite diagonal dimension is always hereditary in that sense, and we give an example where this property does not occur.
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