  In his study of amenable unitary representations, M. E. B. Bekka asked if
there is an analogue for such representations of the remarkable fixed-point
property for amenable groups. In this paper, we prove such a fixed-point
theorem in the more general context of a $G$-amenable von Neumann algebra $M$,
where $G$ is a locally compact group acting on $M$. The F{\o}lner conditions of
Connes and Bekka are extended to the case where $M$ is semifinite and admits a
faithful, semifinite, normal trace which is invariant under the action of $G$.
