  We study bounded operators defined in terms of the regular representations of
the $C^*$-algebra of an amenable, Hausdorff, second countable locally compact
groupoid endowed with a continuous $2$-cocycle. We concentrate on spectral
quantities associated to natural quotients of this twisted algebra, such as the
essential spectrum, the essential numerical range, and Fredholm properties. We
obtain decompositions for the regular representations associated to units of
the groupoid belonging to a free locally closed orbit, in terms of spectral
quantities attached to points (or orbits) in the boundary of this main orbit.
As examples, we discuss various classes of magnetic pseudo-differential
operators on nilpotent groups. We also prove localization and non-propagation
properties associated to suitable parts of the essential spectrum. These are
applied to twisted groupoids having a totally intransitive groupoid restriction
at the boundary.
