Paper detail

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

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 and which do not. 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. As a consequence the Toeplitz algebras of these groups are universal for covariant isometric representations on Hilbert space, and their representations are faithful if the isometries satisfy a properness condition given by Laca and Raeburn. An application of this to right-angled Artin groups gives 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. In contrast, the nonabelian Artin groups of finite type considered by Brieskorn and Saito and Deligne have canonical quasi-lattice orders that are not amenable in the sense of Nica, so their Toeplitz algebras are not universal and the C^*-algebra generated by a collection of isometries satisfying the Artin relations fails to be unique.