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

TL;DR

This paper develops a global pseudo-differential calculus on unimodular type I locally compact groups, linking it with Wigner transforms, Weyl systems, and $C^*$-algebras, with applications to nilpotent Lie groups.

Contribution

It introduces a new pseudo-differential calculus for operator-valued symbols on such groups and explores its connections with Wigner transforms, Weyl systems, and $C^*$-algebras.

Findings

Established a global pseudo-differential calculus on $G \times \hat G$.

Connected the calculus with crossed product $C^*$-algebras.

Applied the framework to spectral analysis on nilpotent Lie groups.

Abstract

Let $G$ be a unimodular type I second countable locally compact group and $\hat G$ its unitary dual. We introduce and study a global pseudo-differential calculus for operator-valued symbols defined on $G\times\hat G$, 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.