Journal of Operator Theory

Volume 36, Issue 2, Fall 1996 pp. 317-334.

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

Authors: Guyan Robertson (1) and Tim Steger (2)
Author institution:(1) Department of Mathematics, University of Newcastle, NSW 2308, AUSTRALIA
(2) Istituto di Matematica e Fisica, Università di Sassari, via Vienna 2, 07100 Sassari, ITALIA

Summary: An $\tilde A_2$ group $\Gamma$ acts simply transitively on the vertices of an affine building $\Delta$. We study certain subgroups $\Gamma_0 \cong \mathbb Z^2$ which act on certain apartments of $\Delta$. 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 Pukánszky invariant contains a type $I_\infty$ summand.

Keywords: Group factor, abelian subalgebra, affine building.

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