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Rokhlin Dimension for Flows

Abstract

We introduce a notion of Rokhlin dimension for one parameter automorphism groups of \({C^*}\)-algebras. This generalizes Kishimoto’s Rokhlin property for flows, and is analogous to the notion of Rokhlin dimension for actions of the integers and other discrete groups introduced by the authors and Zacharias in previous papers. We show that finite nuclear dimension and absorption of a strongly self-absorbing \({C^*}\)-algebra are preserved under forming crossed products by flows with finite Rokhlin dimension, and that these crossed products are stable. Furthermore, we show that a flow on a commutative \({C^*}\)-algebra arising from a free topological flow has finite Rokhlin dimension, whenever the spectrum is a locally compact metrizable space with finite covering dimension. For flows that are both free and minimal, this has strong consequences for the associated crossed product \({C^{*}}\)-algebras: Those containing a non-zero projection are classified by the Elliott invariant (for compact manifolds this consists of topological \({K}\)-theory together with the space of invariant probability measures and a natural pairing given by the Ruelle–Sullivan map).

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Correspondence to Ilan Hirshberg.

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Communicated by Y. Kawahigashi

This research was supported by GIF Grant 1137/2011, SFB 878 Groups, Geometry and Actions and ERC Grant No. 267079. Part of the research was conducted at the Fields institute during the 2014 thematic program on abstract harmonic analysis, Banach and operator algebras, and at the Mittag–Leffler institute during the 2016 program on Classification of Operator Algebras: Complexity, Rigidity, and Dynamics.

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Hirshberg, I., Szabó, G., Winter, W. et al. Rokhlin Dimension for Flows. Commun. Math. Phys. 353, 253–316 (2017). https://doi.org/10.1007/s00220-016-2762-0

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  • DOI: https://doi.org/10.1007/s00220-016-2762-0