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

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

2005-04-18

arxiv.org/abs/math/0504362v1

Mathematics

Operator Algebras

Scientific paper

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 $f\in\ell_2(\Gamma)$ 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 $\Gamma_0\cong\bb Z^r$ be the image of the fundamental group of $T^r$ in $\Gamma$. Then $\vn(\Gamma_0)$ is a maximal abelian $\star$-subalgebra of $\vn(\Gamma)$ and its unitary normalizer is as small as possible. If $M$ has constant negative curvature then the Puk\'anszky invariant of $\vn(\Gamma_0)$ is $\infty$.

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