Abelian subalgebras of von Neumann algebras from flat tori in locally symmetric spaces

摘要

Consider a compact locally symmetric space M of rank r, with fundamental group Gamma. The von Neumann algebra VN(Gamma) is the convolution algebra of functions f is an element of l(2)(Gamma) which act by left convolution on l(2)(Gamma). Let T-r be a totally geodesic flat torus of dimension r in M and let Gamma(o)congruent to Z(r) be the image of the fundamental group of T-r in Gamma. Then VN(Gamma(0)) is a maximal abelian *-subalgebra of VN(Gamma) and its unitary normalizer is as small as possible. If M has constant negative curvature then the Pukanszky invariant of VN(Gamma(0)) is infinity. (C) 2005 Elsevier Inc. All rights reserved.