Maximal abelian subalgebras of the group factor of an $\widetilde A_2$ group

TL;DR

This paper investigates maximal abelian subalgebras within the group von Neumann algebra of an $ ilde{A}_2$ group, revealing their singularity and structure through subgroup actions on affine buildings.

Contribution

It characterizes maximal abelian subalgebras arising from specific subgroup actions in $ ilde{A}_2$ groups and analyzes their properties within the von Neumann algebra framework.

Findings

Subgroups acting simply transitively produce maximal abelian, singular subalgebras.

The Pukánszky invariant includes a type I_infinity component.

The study links geometric actions to operator algebra structures.

Abstract

An $\widetilde A_2$ group $\Gamma$ acts simply transitively on the vertices of an affine building $\triangle$. We study certain subgroups $\Gamma_0 \cong {\Bbb Z}^2$ which act on certain apartments of $\triangle$. If one of these subgroups acts simply transitively on an apartment, then the corresponding subalgebra of the group von Neumann algebra is maximal abelian and singular. Moreover the Puk\'anszky invariant contains a type $I_{\infty}$ summand.