  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 $C^*_r(\mathbb F_2)$
considered by Haagerup and Thorbjornsen.
