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 G0W0 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 G0W0 quasiparticle corrections. However, one-shot G0W0 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 G0W0 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 nontrivial 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 that 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%.