### 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).

Original language | English (US) |
---|---|

Pages (from-to) | 253-316 |

Number of pages | 64 |

Journal | Communications In Mathematical Physics |

Volume | 353 |

Issue number | 1 |

DOIs | |

State | Published - Jul 1 2017 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications In Mathematical Physics*,

*353*(1), 253-316. https://doi.org/10.1007/s00220-016-2762-0

}

*Communications In Mathematical Physics*, vol. 353, no. 1, pp. 253-316. https://doi.org/10.1007/s00220-016-2762-0

**Rokhlin Dimension for Flows.** / Hirshberg, Ilan; Szabó, Gábor; Winter, Wilhelm; Wu, Jianchao.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Rokhlin Dimension for Flows

AU - Hirshberg, Ilan

AU - Szabó, Gábor

AU - Winter, Wilhelm

AU - Wu, Jianchao

PY - 2017/7/1

Y1 - 2017/7/1

N2 - 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).

AB - 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).

UR - http://www.scopus.com/inward/record.url?scp=84992179543&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84992179543&partnerID=8YFLogxK

U2 - 10.1007/s00220-016-2762-0

DO - 10.1007/s00220-016-2762-0

M3 - Article

AN - SCOPUS:84992179543

VL - 353

SP - 253

EP - 316

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 1

ER -