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

Spectral Theory in a Twisted Groupoid Setting: Spectral Decompositions, Localization and Fredholmness