  We construct an action of the Thompson group F on a compact space built from
pairs of infinite, binary rooted trees. The action arises as an F-equivariant
compactification of the action of F by translations on one of its homogeneous
spaces, F/H_2, corresponding to a certain subgroup H_2 of F. The representation
of F on the Hilbert space l^2(F/H_2) is faithful on the complex group algebra
C[F].
