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Amenable and inner amenable actions and approximation properties for crossed products by locally compact groups

Published online by Cambridge University Press:  24 May 2021

Andrew McKee  and
Reyhaneh Pourshahami
Andrew McKee*
Affiliation:
Faculty of Mathematics, University of Białystok, ul. K. Ciołkowskiego 1M, Białystok15-425, Poland
Reyhaneh Pourshahami
Affiliation:
Department of Mathematics, Kharazmi University, 50 Taleghani Avenue, 15618Tehran, Iran e-mail: std_reyhaneh.pourshahami@khu.ac.ir
*
e-mail: amckee240@qub.ac.uk
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Abstract

Amenable actions of locally compact groups on von Neumann algebras are investigated by exploiting the natural module structure of the crossed product over the Fourier algebra of the acting group. The resulting characterization of injectivity for crossed products generalizes a result of Anantharaman-Delaroche on discrete groups. Amenable actions of locally compact groups on $C^*$ -algebras are investigated in the same way, and amenability of the action is related to nuclearity of the corresponding crossed product. A survey is given to show that this notion of amenable action for $C^*$ -algebras satisfies a number of expected properties. A notion of inner amenability for actions of locally compact groups is introduced, and a number of applications are given in the form of averaging arguments, relating approximation properties of crossed product von Neumann algebras to properties of the components of the underlying $w^*$ -dynamical system. We use these results to answer a recent question of Buss, Echterhoff, and Willett.

Keywords

Amenable actioninner amenable actioncrossed productoperator moduleinjectivity
Type
Article
Information
Canadian Mathematical Bulletin , Volume 65 , Issue 2 , June 2022 , pp. 381 - 399
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Copyright
© Canadian Mathematical Society 2021

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