We prove the weak-$*$ convergence of a certain sequence of averages of unitary operators associated to the action of the free group on its Gromov boundary. This result, which can be thought as an ergodic theorem à la von Neumann with coefficients, provides a new proof of the irreducibility of the quasi-regular representation of the free group.
"An ergodic theorem for the quasi-regular representation of the free group." Bull. Belg. Math. Soc. Simon Stevin 24 (2) 243 - 255, april 2017. https://doi.org/10.36045/bbms/1503453708