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On the Toeplitz algebras of right-angled and finite-type Artin groups

Part of: Locally compact groups and their algebras Special aspects of infinite or finite groups Selfadjoint operator algebras Special classes of linear operators

Published online by Cambridge University Press:  09 April 2009

John Crisp  and
Marcelo Laca
John Crisp
Affiliation:
Laboratoire de Topologie, Université de Bourgogne, UMR 5584 du CNRS, B.P.47 870, 21078 Dijon Cedex, France e-mail: crisp@topolog.u-bourgogne.fr
Marcelo Laca
Affiliation:
Department of Mathematics, The University of Newcastle, NSW 2308, Australia
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Abstract

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The graph product of a family of groups lies somewhere 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.

Keywords

graph productquasi-lattice ordercovariant isometric representationToeplitz algebraArtin group

MSC classification

Secondary: 20F36: Braid groups; Artin groups 22D25: $C^*$-algebras and $W^*$-algebras in relation to group representations 46L05: General theory of $C^*$-algebras 46L55: Noncommutative dynamical systems 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 72 , Issue 2 , April 2002 , pp. 223 - 246
Copyright
Copyright © Australian Mathematical Society 2002

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