J. Austral. Math. Soc.
72 (2002), 223245

On the Toeplitz algebras of rightangled and finitetype Artin groups

John Crisp
Laboratoire de Topologie
Université de Bourgogne
UMR 5584 du CNRS
B.P. 47 870
21078 Dijon Cedex
France
crisp@topolog.ubourgogne.fr


and

Marcelo Laca
Department of Mathematics
The University of Newcastle
NSW 2308 Australia
Current address:
Mathematisches Institut
Westfalische WilhelmsUniversitat
Einsteinstr. 62, 48149 Munster
Germany
laca@math.unimuenster.de



Abstract

between their direct and free products, with the
graph determining which pairs of groups commute.
We show that the graph product of quasilattice
ordered groups is quasilattice ordered, and,
when the underlying groups are amenable, that it
satisfies Nica's amenability condition for
quasilattice orders. The associated Toeplitz
algebras have a universal property, and their
representations are faithful if the generating
isometries satisfy a joint properness condition.
When applied to rightangled Artin groups this
yields a uniqueness theorem for the
C^{*}algebra generated by a collection of isometries
such that any two of them either
commute or else have orthogonal ranges. The
analogous result fails to hold for the nonabelian
Artin groups of finite type considered by
Brieskorn and Saito, and Deligne.

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