The boundary of universal discrete quantum groups, exactness, and factoriality

Abstract

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