Let G be a unimodular type I second countable locally compact
group and let Gb be its unitary dual. We introduce and study a global
pseudo-differential calculus for operator-valued symbols defined on G × Gb ,
and its relations to suitably defined Wigner transforms and Weyl systems.
We also unveil its connections with crossed products C
∗
-algebras associated
to certain C
∗
-dynamical systems, and apply it to the spectral analysis
of covariant families of operators. Applications are given to nilpotent
Lie groups, in which case we relate quantizations with operator-valued and
scalar-valued symbols.