  We have implemented the so called GW approximation (GWA) based on an
all-electron full-potential Projector Augmented Wave (PAW) method. For the
screening of the Coulomb interaction W we tested three different plasmon-pole
dielectric function models, and showed that the accuracy of the quasiparticle
energies is not sensitive to the the details of these models. We have then
applied this new method to compute the quasiparticle band structure of some
small, medium and large-band-gap semiconductors: Si, GaAs, AlAs, InP, SiMg$_2$,
C and (insulator) LiCl. A special attention was devoted to the convergence of
the self-energy with respect to both the {\bf k}-points in the Brillouin zone
and to the number of reciprocal space $\bf G$-vectors. The most important
result is that although the all-electron GWA improves considerably the
quasiparticle band structure of semiconductors, it does not always provide the
correct energy band gaps as originally claimed by GWA pseudopotential type of
calculations. We argue that the decoupling between the valence and core
electrons is a problem, and is some what hidden in a pseudopotential type of
approach.
