On the Toeplitz C*-algebras of Artin Groups
by
Marcelo Laca
University of Newcastle, Australia
Coauthors: John Crisp (University of Southampton, U.K.)

The graph product of a family of quasi-lattice ordered groups lies somewhere between their direct and free products and is quasi-lattice ordered. Following a strategy of Laca and Raeburn, we show that if the underlying groups are amenable, then their graph product satisfies an amenability condition of Nica. As a consequence, their Toeplitz C*-algebras are universal for covariant isometric representations, and faithfulness of a representation depends only on a mild nonsingularity condition. An application of this to right-angled Artin groups gives a uniqueness theorem for the C*-algebra generated by a collection of isometries having the property that any two of them either commute and *-commute or else have orthogonal ranges. This unifies and generalizes several celebrated results about uniqueness of C*-algebras generated by semigroups of isometries due to Coburn, Douglas, Murphy and Cuntz. On the negative side, the nonabelian Artin groups of finite type considered by Brieskorn and Saito have canonical quasi-lattice orders that are not amenable in the sense of Nica, so their Toeplitz algebras are not universal, nonsingularity of a representation does not entail its faithfulness, and the C*-algebra generated by a collection of isometries satisfying the Artin relations fails to be unique.

Date received: March 29, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacw-16.