Rocky Mountain Journal of Mathematics

Invariant means and property $T$ of crossed products

Qing Meng and Chi-Keung Ng

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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.

Article information

Rocky Mountain J. Math., Volume 48, Number 3 (2018), 905-912.

First available in Project Euclid: 2 August 2018

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Zentralblatt MATH identifier

Primary: 37A15: General groups of measure-preserving transformations [See mainly 22Fxx] 37A25: Ergodicity, mixing, rates of mixing 46L05: General theory of $C^*$-algebras 46L55: Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20]

$C^*$-crossed products property $T$ strong property $T$


Meng, Qing; Ng, Chi-Keung. Invariant means and property $T$ of crossed products. Rocky Mountain J. Math. 48 (2018), no. 3, 905--912. doi:10.1216/RMJ-2018-48-3-905.

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