On relative commutants of subalgebras in group and tracial crossed product von Neumann algebras

Let be a discrete group acting on a compact Hausdorff space X.Given x X and Prob(X), we introduce the notion of contraction of towards x with respect to unitary elements of a group von Neumann algebra not necessarily coming from group elements.Using this notion, we study relative commutants of subalgebras in tracial crossed product von Neumann algebras.The results are applied to negatively curved groups and SL(d, ) for d 2.