What is a quantum cluster C*-algebra?

Quantum cluster algebras are algebras with a rich combinatorial structure and remarkable properties. In recent years, the cluster structures on various quantum groups have been used to construct so-called positive representations. We will first describe the Fock and Goncharov realization of cluster variables as positive operators on a Hilbert space, and how to implement mutation at this level. Afterwards we will present ongoing work with K. De Commer and G. Janssens on how to then make a C*-algebra generated by these cluster variables, and how this relates to positive representations.