Group amenability properties for von Neumann algebras Anthony LauAlan Paterson 22D1022D2543A07amenable representation$G$-amenability$G$-fixed point propertyHopf-von Neumann algebrasF\o lner conditions 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 $G$. Indiana University Mathematics Journal 2006 text pdf 10.1512/iumj.2006.55.2787 10.1512/iumj.2006.55.2787 en Indiana Univ. Math. J. 55 (2006) 1363 - 1388 state-of-the-art mathematics http://iumj.org/access/