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 $f\in\ell_2(Γ)$ which act by left convolution on $\ell_2(Γ)$. Let $T^r$ be a totally geodesic flat torus of dimension $r$ in $M$ and let $Γ_0\cong\bb Z^r$ be the image of the fundamental group of $T^r$ in $Γ$. Then $\vn(Γ_0)$ is a maximal abelian $\star$-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 $\infty$.