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2018 Invariant means and property $T$ of crossed products
Qing Meng, Chi-Keung Ng
Rocky Mountain J. Math. 48(3): 905-912 (2018). DOI: 10.1216/RMJ-2018-48-3-905

Abstract

Let $\Gamma $ be a discrete group that acts on a semi-finite measure space $(\Omega , \mu )$ such that there is no $\Gamma $-invariant function in $L^1(\Omega , \mu )$. We show that the absence of the $\Gamma $-invariant mean on $L^\infty (\Omega ,\mu )$ is equivalent to the property $T$ of the reduced $C^*$-crossed product of $L^\infty (\Omega ,\mu )$ by $\Gamma $. In particular, if $\Lambda $ is a countable group acting ergodically on an infinite $\sigma $-finite measure space $(\Omega , \mu )$, then there exists a $\Lambda $-invariant mean on $L^\infty (\Omega , \mu )$ if and only if the corresponding crossed product does not have property $T$. Moreover, if $\Gamma $ is an ICC group, then $\Gamma $ is inner amenable if and only if $\ell ^\infty (\Gamma \setminus \{e\})\rtimes _{\mathbf {i},r} \Gamma $ does not have property $T$, where $\mathbf {i}$ is the conjugate action. On the other hand, a non-compact locally compact group $G$ is amenable if and only if $L^\infty (G)\rtimes _{\mathbf {lt}, r} G_\mathrm {d}$ does not have property $T$, where $G_\mathrm {d}$ is the group $G$ equipped with the discrete topology and $\mathbf {lt}$ is the left translation.

Citation

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Qing Meng. Chi-Keung Ng. "Invariant means and property $T$ of crossed products." Rocky Mountain J. Math. 48 (3) 905 - 912, 2018. https://doi.org/10.1216/RMJ-2018-48-3-905

Information

Published: 2018
First available in Project Euclid: 2 August 2018

zbMATH: 06917353
MathSciNet: MR3835578
Digital Object Identifier: 10.1216/RMJ-2018-48-3-905

Subjects:
Primary: 37A15, 37A25, 46L05, 46L55

Rights: Copyright © 2018 Rocky Mountain Mathematics Consortium

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Vol.48 • No. 3 • 2018
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