Published online by Cambridge University Press: 20 November 2018
It is well known that a discrete group that is both amenable and has Kazhdan’s Property $T$ must be finite. In this note we generalize this statement to the case of transformation groups. We show that if
$G$ is a discrete amenable group acting on a compact Hausdorff space
$X$, then the transformation group
${{C}^{*}}$-algebra
${{C}^{*}}\left( X,\,G \right)$ has Property
$T$ if and only if both
$X$ and
$G$ are finite. Our approach does not rely on the use of tracial states on
${{C}^{*}}\left( X,\,G \right)$.