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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Comparison and pure infiniteness of crossed products


Author: Xin Ma
Journal: Trans. Amer. Math. Soc. 372 (2019), 7497-7520
MSC (2010): Primary 37B05, 46L35
DOI: https://doi.org/10.1090/tran/7927
Published electronically: August 28, 2019
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Abstract: Let $ \alpha : G\curvearrowright X$ be a continuous action of an infinite countable group on a compact Hausdorff space. We show that, under the hypothesis that the action $ \alpha $ is topologically free and has no $ G$-invariant regular Borel probability measure on $ X$, dynamical comparison implies that the reduced crossed product of $ \alpha $ is purely infinite and simple. This result, as an application, shows a dichotomy between stable finiteness and pure infiniteness for reduced crossed products arising from actions satisfying dynamical comparison. We also introduce the concepts of paradoxical comparison and the uniform tower property. Under the hypothesis that the action $ \alpha $ is exact and essentially free, we show that paradoxical comparison together with the uniform tower property implies that the reduced crossed product of $ \alpha $ is purely infinite. As applications, we provide new results on pure infiniteness of reduced crossed products in which the underlying spaces are not necessarily zero dimensional. Finally, we study the type semigroups of actions on the Cantor set in order to establish the equivalence of almost unperforation of the type semigroup and comparison. This sheds light on a question arising in a paper of Rørdam and Sierakowski.


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Additional Information

Xin Ma
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: dongodel@math.tamu.edu

DOI: https://doi.org/10.1090/tran/7927
Received by editor(s): September 12, 2018
Received by editor(s) in revised form: June 3, 2019
Published electronically: August 28, 2019
Article copyright: © Copyright 2019 American Mathematical Society