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Rokhlin dimension for flows. (English) Zbl 1383.46051

In the paper under review, the authors introduce and study a notion of Rokhlin dimension for actions of the reals (flows) on separable \(C^*\)-algebras, generalizing Kishimoto’s Rokhlin property for flows [A. Kishimoto, Commun. Math. Phys. 179, No. 3, 599–622 (1996; Zbl 0853.46067)]. As in other instances where Rokhlin dimension has been defined (for example, for residually finite groups by G. Szabó et al. [“Rokhlin dimension for actions of residually finite groups”, Ergod. Theory Dyn. Syst. 1–57 (2017; doi:10.1017/etds.2017.113] or for compact groups by the reviewer [Indiana Univ. Math. J. 66, No. 2, 659–703 (2017; Zbl 1379.46053)]), there is a related and stronger dimension, called “Rokhlin dimension with commuting towers”, where some additional commuting conditions are imposed. (While these notions are known to be different for groups with torsion, it is unclear whether they differ for flows.)
The main results of the paper refer to the crossed products by such actions. First, if the flow has finite Rokhlin dimension, then the crossed product is automatically stable and its nuclear dimension is finite whenever that of the coefficient algebra is. Second, if the flow has finite Rokhlin dimension with commuting towers, then absorption of a strongly self-absorbing \(C^*\)-algebra passes from the coefficient algebra to the crossed product.
Using very powerful results of A. Bartels et al. [Geom. Topol. 12, No. 3, 1799–1882 (2008; Zbl 1185.20045)], developed in the context of the Farrell-Jones conjecture, the authors show that free flows on finite-dimensional compact, metrizable spaces have finite Rokhlin dimension. It follows that crossed products of free and minimal flows on such spaces are classifiable by their Elliott invariant as long as they contain a nontrivial projection.

MSC:

46L55 Noncommutative dynamical systems
54H20 Topological dynamics (MSC2010)
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