论文标题
$ c^*$ - 代数有限复杂性
$C^*$-algebras with finite complexity
论文作者
论文摘要
$ C^*$ - 代数的复杂性排名由第二作者和YU引入UCT：非常大致，如果您可以反复将$ c^*$ - 代数最多切成$ n $ times，最终以有限的尺寸降低。在本文中，我们研究复杂性等级，也是我们引入的弱复杂性等级。较弱的复杂性最多可以被认为是“两色局部有限维度”。 我们首先表明，对于可分离，联合和简单的$ c^*$ - 代数，弱复杂性等级等同于核维度一号的结合和实际等级零。特别是，这表明所有核$ c^*$ - 代数的UCT相当于弱复杂性等级的平等，而Kirchberg代数的复杂性等级为零$ K $ - 理论组。但是，我们还显示了使用$ K $的理论障碍（$ k_1 $中的扭转）表明，弱复杂性排名第一，而复杂性排名一般并不相同。 然后，我们使用Kirchberg-Phillips分类定理来计算所有UCT Kirchberg代数的复杂性等级：它始终是一个或两个，当$ K_1 $ -Group是无扭力的情况下，等级的情况下是一个级别的情况。
Complexity rank for $C^*$-algebras was introduced by the second author and Yu for applications towards the UCT: very roughly, this rank is at most $n$ if you can repeatedly cut the $C^*$-algebra in half at most $n$ times, and end up with something finite dimensional. In this paper, we study complexity rank, and also a weak complexity rank that we introduce; having weak complexity rank at most one can be thought of as `two-colored local finite-dimensionality'. We first show that for separable, unital, and simple $C^*$-algebras, weak complexity rank one is equivalent to the conjunction of nuclear dimension one and real rank zero. In particular, this shows that the UCT for all nuclear $C^*$-algebras is equivalent to equality of the weak complexity rank and the complexity ranks for Kirchberg algebras with zero $K$-theory groups. However, we also show using a $K$-theoretic obstruction (torsion in $K_1$) that weak complexity rank one and complexity rank one are not the same in general. We then use the Kirchberg-Phillips classification theorem to compute the complexity rank of all UCT Kirchberg algebras: it is always one or two, with the rank one case occurring if and only if the $K_1$-group is torsion free.
