Approximation properties for groups and von Neumann algebras.

Abstract: This talk is about recent advances concerning approximation properties for groups and group von Neumann algebras. In 1994 Jon Kraus and I introduced a new approximation property (AP) for locally compact groups and we proved that for dicrete groups AP is equivalent to the property W*-OAP of Effros and Ruan for the group von Neumann algebra. In 2011 Vincent Lafforgue and Michael de la Salle has proved that SL(n,R) and SL(n,Z) does not have the property AP for n >= 3. In a joint work with Tim de Laat (Duke Math. J. 2012), we extend their result by proving that Sp(2,R) and more generally all simple connected Lie groups of real rank >=2 and with finite center do not have the AP. The proof uses some careful estimates of Jacobi polynomials obtained in collaboration with Henrik Shlichtkrull. In a second paper (math arXiv 2013) Tim de Laat and I have now removed the finite center condition from our result in Duke.