Pseudo-differential operators, Wigner transform and Weyl systems on type I locally compact groups

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