  We present a formalism for strongly correlated electrons systems which
consists in a local approximation of the dynamical three-leg interaction
vertex. This vertex is self-consistently computed with a quantum impurity model
with dynamical interactions in the charge and spin channels, similar to
dynamical mean field theory (DMFT) approaches. The electronic self-energy and
the polarization are both frequency and momentum dependent. The method
interpolates between the spin-fluctuation or GW approximations at weak coupling
and the atomic limit at strong coupling. We apply the formalism to the Hubbard
model on a two-dimensional square lattice and show that as interactions are
increased towards the Mott insulating state, the local vertex acquires a strong
frequency dependence, driving the system to a Mott transition, while at low
enough temperatures the momentum-dependence of the self-energy is enhanced due
to large spin fluctuations. Upon doping, we find a Fermi arc in the
one-particle spectral function, which is one signature of the pseudo-gap state.
