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

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