Duke Mathematical Journal vol:140 issue:1 pages:35-84
We study the C*-algebras and von Neumann algebras associated with the universal
discrete quantum groups. They give rise to full prime factors and simple exact C*-
algebras. The main tool in our work is the study of an amenable boundary action,
yielding the Akemann-Ostrand property. Finally, this boundary can be identified with
the Martin or the Poisson boundary of a quantum random walk.