# On C*-completions of discrete quantum group rings

## TL;DR

This paper investigates the conditions under which C*-completions of discrete quantum group rings are unique, revealing that duals of q-deformed Lie groups are not C*-unique, while some non-locally finite quantum groups are.

## Contribution

It establishes a connection between just infiniteness, C*-uniqueness, and the structure of discrete quantum groups, providing new examples and counterexamples.

## Key findings

- Duals of q-deformations of simply connected semisimple Lie groups are not C*-unique.
- An example of a non-locally finite discrete quantum group that is C*-unique.
- Relation between just infiniteness of C*-algebras and C*-uniqueness of quantum groups.

## Abstract

We discuss just infiniteness of C*-algebras associated to discrete quantum groups and relate it to the C*-uniqueness of the quantum groups in question, i.e. to the uniqueness of a C*-completion of the underlying Hopf *-algebra. It is shown that duals of q-deformations of simply connected semisimple compact Lie groups are never C*-unique. On the other hand we present an example of a discrete quantum group which is not locally finite and yet is C*-unique.