J. Austral. Math. Soc.  72 (2002), 223-245
On the Toeplitz algebras of right-angled and finite-type Artin groups

John Crisp
  Laboratoire de Topologie
  Université de Bourgogne
  UMR 5584 du CNRS
  B.P. 47 870
  21078 Dijon Cedex
Marcelo Laca
  Department of Mathematics
  The University of Newcastle
  NSW 2308 Australia
  Current address:
Mathematisches Institut
  Westfalische Wilhelms-Universitat
  Einsteinstr. 62, 48149 Munster

between their direct and free products, with the graph determining which pairs of groups commute. We show that the graph product of quasi-lattice ordered groups is quasi-lattice ordered, and, when the underlying groups are amenable, that it satisfies Nica's amenability condition for quasi-lattice 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 right-angled 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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