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

Guyan Robertson

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

Guyan Robertson

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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 $\infty$.

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BibTeX RIS

Guyan Robertson. 2005-04-18. Abelian subalgebras of von Neumann algebras from flat tori in locally symmetric spaces. https://arxiv.org/abs/math/0504362

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