Rokhlin dimension: obstructions and permanence properties
By Ilan Hirshberg, N. Christopher Phillips. Documenta Mathematica, 2014
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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
Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics
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Citations in the corpus
- Rokhlin dimension for actions of residually finite groups (2014, methodology): “For a selection of other papers that, at least in large part, emerged out of the ideas within [28], see [2, 7, 16, 17, 25, 26, 29, 44, 45, 71].”
- Rokhlin dimension for compact group actions (2014, methodology): “The paper [13] consists of a further study of finite Rokhlin dimension, where the authors extend the notion to the non-unital setting, and also derive some Ktheoretical obstructions in the commuting tower version.”