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The boundary of universal discrete quantum groups, exactness, and factoriality. (English) Zbl 1129.46062
Summary: 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.

MSC:
46L65 Quantizations, deformations for selfadjoint operator algebras
46L55 Noncommutative dynamical systems
46L35 Classifications of \(C^*\)-algebras
46L53 Noncommutative probability and statistics
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