Invariant means and property $T$ of crossed products Article Swipe
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· 2018
· Open Access
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· DOI: https://doi.org/10.1216/rmj-2018-48-3-905
· OA: W2886805948
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.
