Asymptotic representations of the reduced C*-algebra of a free group: an example

We give an example of a non-trivial asymptotic representation of the reduced C*-algebra of a free group. This example allows to evaluate the asymptotic tensor C*-norm of some elements in tensor product C*-algebras and to show semi-invertibility of the non-invertible extension of $C_r^{\ast }({\mathbb{F}}_2)$ considered by Haagerup and Thorbjornsen.