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 $\Gamma$. The von Neumann algebra $\vn(Γ)$ is the convolution algebra of functions $f\in {\ell }_2(\Gamma )$ which act by left convolution on ${\ell }_2(\Gamma )$. 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 $\Gamma$. 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 $\infty$.

References in corpus (1)

• Ergodic theory and maximal abelian subalgebras of the hyperfinite factor