Invariant means and property $T$ of crossed products

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.