Pseudo-Differential Operators, Wigner Transform and Weyl Systems on Type I Locally Compact Groups

Abstract

Let $\mathsf{G}$ be a unimodular type I second countable locally compact group and let $\hat{\mathsf{G}}$ be its unitary dual. We introduce and study a global pseudo-differential calculus for operator-valued symbols defined on $\mathsf{G}\times \hat{\mathsf{G}}$, and its relations to suitably defined Wigner transforms and Weyl systems. We also unveil its connections with crossed products $C^{\ast }$-algebras associated to certain $C^{\ast }$-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.