CLASSIFIABLE C∗-ALGEBRAS FROM MINIMAL Z-ACTIONS AND THEIR ORBIT-BREAKING SUBALGEBRAS
CLASSIFIABLE C∗-ALGEBRAS FROM MINIMAL Z-ACTIONS AND THEIR ORBIT-BREAKING SUBALGEBRAS
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发表时间：
2020
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通讯作者：
I. Putnam;Karen R. Strung
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作者：
I. Putnam;Karen R. Strung
本文讨论了由极小动力系统产生的C * -代数的k理论可以产生哪些阿贝尔群的问题。利用整数作用有限生成的k理论完整地刻画了空间X的交叉积的k理论，并证明了极小同胚的交叉积用尽了这些可能的k理论的范围。此外，我们可以安排所涉及的最小系统是唯一遍历的，因此它们的C * -代数由它们的艾略特不变量分类。我们还研究了k理论和轨道破断代数的Elliott不变量。我们证明了给定任意可数阿贝群G0和G1以及任意具有有限多个极值点的Choquet单形∆，我们可以找到一个极小的破轨关系，使得相关的C * -代数具有由这对群给出的k理论，并且迹迹状态空间仿射同胚于∆。我们也改进了第二作者之前的结果，使用我们的轨道破缺构造对C * -代数的最小可服从等价关系，实秩为零，在K0和K1中都允许扭转。这些结果对于C * -代数的Elliott分类程序有重要的应用。特别地，我们在确定与栅格等价关系相关的C * -代数的艾略特不变量的范围方面迈出了一步。
In this paper we consider the question of what abelian groups can arise as the K-theory of C∗-algebras arising from minimal dynamical systems. We completely characterize the K-theory of the crossed product of a space X with finitely generated K-theory by an action of the integers and show that crossed products by a minimal homeomorphisms exhaust the range of these possible K-theories. Moreover, we may arrange that the minimal systems involved are uniquely ergodic, so that their C∗-algebras are classified by their Elliott invariants. We also investigate the K-theory and the Elliott invariants of orbit-breaking algebras. We show that given arbitrary countable abelian groups G0 and G1 and any Choquet simplex ∆ with finitely many extreme points, we can find a minimal orbit-breaking relation such that the associated C∗-algebra has K-theory given by this pair of groups and tracial state space affinely homeomorphic to ∆. We also improve on the second author’s previous results by using our orbit-breaking construction to C∗-algebras of minimal amenable equivalence relations with real rank zero that allow torsion in both K0 and K1. These results have important applications to the Elliott classification program for C∗-algebras. In particular, we make a step towards determining the range of the Elliott invariant of the C∗-algebras associated to étale equivalence relations.