TY - JOUR

T1 - Rokhlin Dimension and C*-Dynamics

AU - Hirshberg, Ilan

AU - Winter, Wilhelm

AU - Zacharias, Joachim

N1 - Funding Information:
Research partially supported by the Israel Science Foundation (Grant No. 1471/07), by GIF (Grant No. 1137-30.6/2011), by EPSRC (Grants No. EP/G014019/1 and No. EP/I019227/1), by the DFG (SFB 878) and by the CRM Barcelona.
Publisher Copyright:
© 2015, The Author(s).

PY - 2015/4/1

Y1 - 2015/4/1

N2 - We develop the concept of Rokhlin dimension for integer and for finite group actions on C*-algebras. Our notion generalizes the so-called Rokhlin property, which can be thought of as Rokhlin dimension 0. We show that finite Rokhlin dimension is prevalent and appears in cases in which the Rokhlin property cannot be expected: the property of having finite Rokhlin dimension is generic for automorphisms of Z-stable C*-algebras, where Z denotes the Jiang–Su algebra. Moreover, crossed products by automorphisms with finite Rokhlin dimension preserve the property of having finite nuclear dimension, and under a mild additional hypothesis also preserve Z -stability. In topological dynamics our notion may be interpreted as a topological version of the classical Rokhlin lemma: automorphisms arising from minimal homeomorphisms of finite dimensional compact metrizable spaces always have finite Rokhlin dimension. The latter result has by now been generalized by Szabó to the case of free and aperiodic Z-actions on compact metrizable and finite dimensional spaces.

AB - We develop the concept of Rokhlin dimension for integer and for finite group actions on C*-algebras. Our notion generalizes the so-called Rokhlin property, which can be thought of as Rokhlin dimension 0. We show that finite Rokhlin dimension is prevalent and appears in cases in which the Rokhlin property cannot be expected: the property of having finite Rokhlin dimension is generic for automorphisms of Z-stable C*-algebras, where Z denotes the Jiang–Su algebra. Moreover, crossed products by automorphisms with finite Rokhlin dimension preserve the property of having finite nuclear dimension, and under a mild additional hypothesis also preserve Z -stability. In topological dynamics our notion may be interpreted as a topological version of the classical Rokhlin lemma: automorphisms arising from minimal homeomorphisms of finite dimensional compact metrizable spaces always have finite Rokhlin dimension. The latter result has by now been generalized by Szabó to the case of free and aperiodic Z-actions on compact metrizable and finite dimensional spaces.

UR - http://www.scopus.com/inward/record.url?scp=84925484330&partnerID=8YFLogxK

U2 - 10.1007/s00220-014-2264-x

DO - 10.1007/s00220-014-2264-x

M3 - Article

AN - SCOPUS:84925484330

VL - 335

SP - 637

EP - 670

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 2

ER -