TY - JOUR

T1 - Partial C∗-dynamics and Rokhlin dimension

AU - Abadie, Fernando

AU - Gardella, Eusebio

AU - Geffen, Shirly

N1 - Publisher Copyright:
© 2022 Cambridge University Press. All rights reserved.

PY - 2022/10/3

Y1 - 2022/10/3

N2 - We develop the notion of the Rokhlin dimension for partial actions of finite groups, extending the well-established theory for global systems. The partial setting exhibits phenomena that cannot be expected for global actions, usually stemming from the fact that virtually all averaging arguments for finite group actions completely break down for partial systems. For example, fixed point algebras and crossed products are not in general Morita equivalent, and there is in general no local approximation of the crossed product by matrices over A. Using decomposition arguments for partial actions of finite groups, we show that a number of structural properties are preserved by formation of crossed products, including finite stable rank, finite nuclear dimension, and absorption of a strongly self-Absorbing-Algebra. Some of our results are new even in the global case. We also study the Rokhlin dimension of globalizable actions: while in general it differs from the Rokhlin dimension of its globalization, we show that they agree if the coefficient algebra is unital. For topological partial actions on spaces of finite covering dimension, we show that finiteness of the Rokhlin dimension is equivalent to freeness, thus providing a large class of examples to which our theory applies.

AB - We develop the notion of the Rokhlin dimension for partial actions of finite groups, extending the well-established theory for global systems. The partial setting exhibits phenomena that cannot be expected for global actions, usually stemming from the fact that virtually all averaging arguments for finite group actions completely break down for partial systems. For example, fixed point algebras and crossed products are not in general Morita equivalent, and there is in general no local approximation of the crossed product by matrices over A. Using decomposition arguments for partial actions of finite groups, we show that a number of structural properties are preserved by formation of crossed products, including finite stable rank, finite nuclear dimension, and absorption of a strongly self-Absorbing-Algebra. Some of our results are new even in the global case. We also study the Rokhlin dimension of globalizable actions: while in general it differs from the Rokhlin dimension of its globalization, we show that they agree if the coefficient algebra is unital. For topological partial actions on spaces of finite covering dimension, we show that finiteness of the Rokhlin dimension is equivalent to freeness, thus providing a large class of examples to which our theory applies.

UR - http://www.scopus.com/inward/record.url?scp=85111070837&partnerID=8YFLogxK

U2 - 10.1017/etds.2021.82

DO - 10.1017/etds.2021.82

M3 - Article

AN - SCOPUS:85111070837

SN - 0143-3857

VL - 42

SP - 2991

EP - 3024

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

IS - 10

ER -