TY - JOUR

T1 - Rokhlin dimension

T2 - Duality, tracial properties, and crossed products

AU - Gardella, Eusebio

AU - Hirshberg, Ilan

AU - Santiago, Luis

N1 - Funding Information:
Acknowledgements. The authors would like to thank N. Christopher Phillips for sharing with them some unpublished material that led to Theorem 3.1. The first named author thanks Etienne Blanchard, Marius Dadarlat and Martino Lupini for helpful correspondence, and Hannes Thiel for calling our attention to the references [52] and [66]. We particularly thank Shirly Geffen for thoroughly reading this paper, and for pointing out numerous typos and mistakes. Parts of this work were completed while the authors were attending the following focus semesters: the Thematic Program on Abstract Harmonic Analysis, Banach and Operator Algebras at the Fields Institute, Toronto, Canada, in January–June 2014; the Focus Programme on C∗-algebras at the Universität Münster, Münster, Germany, in April–July 2015; and Classification of Operator Algebras: Complexity, Rigidity, and Dynamics at the Institut Mittag-Leffler, Sweden, in January– April 2016. The financial support provided by these programs is gratefully acknowledged. Further work was conducted during visits of the first named author to Aberdeen in May 2015, and to the second named author in October 2016, and of the second named author to Münster in June 2015. The authors thank the hosting math departments for their hospitality. The first named author was partially supported by the D. K. Harrison Prize from the University of Oregon, by the Deutsche Forschungsgemeinschaft (SFB 878), and by a Postdoctoral Research Fellowship from the Humboldt Foundation. This research was supported by GIF grant 1137/2011, and by Israel Science Foundation Grant 476/16.
Publisher Copyright:
© 2019 The Author(s).

PY - 2021/2/1

Y1 - 2021/2/1

N2 - We study compact group actions with finite Rokhlin dimension, particularly in relation to crossed products. For example, we characterize the duals of such actions, generalizing previous partial results for the Rokhlin property. As an application, we determine the ideal structure of their crossed products. Under the assumption of so-called commuting towers, we show that taking crossed products by such actions preserves a number of relevant classes of C*-algebras, including:-absorbing C*-algebras, where is a strongly self-absorbing C*-algebra; stable C*-algebras; C*-algebras with finite nuclear dimension (or decomposition rank); C*-algebras with finite stable rank (or real rank); and C*-algebras whose-theory is either trivial, rational, or n-divisible for n∈N. The combination of nuclearity and the universal coefficient theorem (UCT) is also shown to be preserved by these actions. Some of these results are new even in the well-studied case of the Rokhlin property. Additionally, and under some technical assumptions, we show that finite Rokhlin dimension with commuting towers implies the (weak) tracial Rokhlin property. At the core of our arguments is a certain local approximation of the crossed product by a continuous C(X)-algebra with fibers that are stably isomorphic to the underlying algebra. The space is computed in some cases of interest, and we use its description to construct a Z2-action on a unital AF-algebra and on a unital Kirchberg algebra satisfying the UCT, whose Rokhlin dimensions with and without commuting towers are finite but do not agree.

AB - We study compact group actions with finite Rokhlin dimension, particularly in relation to crossed products. For example, we characterize the duals of such actions, generalizing previous partial results for the Rokhlin property. As an application, we determine the ideal structure of their crossed products. Under the assumption of so-called commuting towers, we show that taking crossed products by such actions preserves a number of relevant classes of C*-algebras, including:-absorbing C*-algebras, where is a strongly self-absorbing C*-algebra; stable C*-algebras; C*-algebras with finite nuclear dimension (or decomposition rank); C*-algebras with finite stable rank (or real rank); and C*-algebras whose-theory is either trivial, rational, or n-divisible for n∈N. The combination of nuclearity and the universal coefficient theorem (UCT) is also shown to be preserved by these actions. Some of these results are new even in the well-studied case of the Rokhlin property. Additionally, and under some technical assumptions, we show that finite Rokhlin dimension with commuting towers implies the (weak) tracial Rokhlin property. At the core of our arguments is a certain local approximation of the crossed product by a continuous C(X)-algebra with fibers that are stably isomorphic to the underlying algebra. The space is computed in some cases of interest, and we use its description to construct a Z2-action on a unital AF-algebra and on a unital Kirchberg algebra satisfying the UCT, whose Rokhlin dimensions with and without commuting towers are finite but do not agree.

KW - -algebra

KW - C

KW - Rokhlin dimension

KW - crossed products

UR - http://www.scopus.com/inward/record.url?scp=85074167685&partnerID=8YFLogxK

U2 - 10.1017/etds.2019.68

DO - 10.1017/etds.2019.68

M3 - Article

AN - SCOPUS:85074167685

SN - 0143-3857

VL - 41

SP - 408

EP - 460

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

IS - 2

ER -