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

Abstract

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