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