  We calculate the electronic structure of the narrow gap semiconductors PbTe,
SnTe and GeTe in the cubic phase using density functional theory (DFT) and the
$G_0W_0$ method. Within DFT, we show that the band ordering obtained with a
conventional semilocal exchange-correlation approximation is correct for SnTe
and GeTe but wrong for PbTe. The correct band ordering at the high-symmetry
point L is recovered adding $G_0W_0$ quasiparticle corrections. However,
one-shot $G_0W_0$ produces artifacts in the band structure due to the wrong
orbital character of the DFT single-particle states at the band edges close to
L. We show that in order to correct these artifacts it is enough to consider
the off-diagonal elements of the $G_0W_0$ self-energy corresponding to these
states. We also investigate the pressure dependence of the band gap for these
materials and the possibility of a transition from a trivial to a non-trivial
topology of the band structure. For PbTe, we predict the band crossover and
topological transition to occur at around 4.8 GPa. For GeTe, we estimate the
topological transition to occur at 1.9 GPa in the constrained cubic phase, a
pressure lower than the one of the structural phase transition from rombohedral
to cubic. SnTe is a crystalline topological insulator at ambient pressure, and
the transition into a trivial topology would take place under a volume
expansion of approximately $10\%$.
