Rokhlin dimension: obstructions and permanence properties
Rokhlin dimension: obstructions and permanence properties
复制标题
罗克林维度：障碍物和永久性属性
DOI：
10.4171/dm/489
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发表时间：
2014
影响因子：
0.9
通讯作者：
N. Phillips
中科院分区：
文献类型：
--
作者：
Ilan Hirshberg;N. Phillips
本文是对第一作者Winter和Zacharias关于有限群作用的有限Rokhlin维数和C-代数上整数的有限Rokhlin维数的进一步研究。我们将有限Rokhlin维数的定义推广到非幺正情形。这个定义对于扩张表现得很好，并且足以建立有限核维数和Z-吸收的持久性。我们建立K理论障碍的存在有限群的行动与有限的Rokhlin维数（在交换塔版本）。特别地，我们证明了在这种意义下，任何非平凡有限群在Jiang-Su代数或Cuntz代数O∞上不存在有限Rokhlin维数的作用. 2010年数学学科分类：46 L55 1 C-代数上的群作用及其相关的交叉积的研究一直是算子代数的一个中心研究课题.人们希望确定群作用的性质，这些性质一方面是常见的，并且足够自然地发生以引起人们的兴趣，另一方面是足够强以用于导出作用或交叉积的有趣性质。满足这些标准的群作用的重要性质的例子是罗克林性质的各种形式，它们在理论的早期就出现了。例如，参见（Izu 01）和其中关于Z的作用的参考文献，以及（Izu 04 a，Izu 04 b，Phi 09，OP 12）对于有限群的情况。单自同构情形的Rokhlin性质是相当普遍的，在某些情况下是通用的，在自同构群中形成了一个稠密的G集（参见（HWZ 15））。但
This paper is a further study of finite Rokhlin dimension for actions of finite groups and the integers on C ∗ -algebras, intro- duced by the first author, Winter, and Zacharias. We extend the definition of finite Rokhlin dimension to the nonunital case. This def- inition behaves well with respect to extensions, and is sufficient to establish permanence of finite nuclear dimension and Z-absorption. We establish K-theoretic obstructions to the existence of actions of finite groups with finite Rokhlin dimension (in the commuting tower version). In particular, we show that there are no actions of any non- trivial finite group on the Jiang-Su algebra or on the Cuntz algebra O∞ with finite Rokhlin dimension in this sense. 2010 Mathematics Subject Classification: 46L55 1 The study of group actions on C ∗ -algebras, and their associated crossed prod- ucts, has always been a central research theme in operator algebras. One would like to identify properties of group actions which on the one hand occur com- monly and naturally enough to be of interest, and on the other hand are strong enough to be used to derive interesting properties of the action or of the crossed product. Examples of important properties for a group action meeting these criteria are the various forms of the Rokhlin property, which arose early on in the theory. See, for instance, (Izu01) and references therein for actions of Z and (Izu04a, Izu04b, Phi09, OP12) for the finite group case. The Rokhlin property for the single automorphism case is quite prevalent, and generic in some cases, forming a dense Gset in the automorphism group (see (HWZ15)). However, it