  We provide a new type of proof for known or new Gohberg lemmas for
pseudodifferential operators on Abelian locally compact groups $\mathrm{X}$. We
use $C^*$-algebraic techniques, which also give spectral results to which the
Gohberg lemma is just a corollary. These results extend most of those appearing
in the literature in various directions. In particular, compactness or a a Lie
structure are not needed. The ideal of all the compact operators in
$\mathrm{L}^2(\mathrm{X})$ is replaced by all the ideals having a crossed
product structure, which is a consistent generalization. We also indicate
several new examples, mostly connected to specific behaviors of functions on
the dual $\Xi$ of $\mathrm{X}$.
