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 (G) over cap be its unitary dual. We introduce and study a global pseudo-differential calculus for operator-valued symbols defined on G x (G) over cap, 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.