Pseudo-differential operators associated to general type I locally compact groups

TL;DR

This paper extends the pseudo-differential calculus from unimodular to all type I locally compact groups, utilizing Plancherel's theorem and connecting with $C^*$-algebras, including explicit examples for affine groups.

Contribution

It generalizes the pseudo-differential calculus to arbitrary type I groups, broadening the scope beyond unimodular cases with new methods and explicit constructions.

Findings

Extended calculus to non-unimodular groups

Connected pseudo-differential operators with $C^*$-algebraic formalism

Provided explicit examples for affine transformation groups

Abstract

In a recent paper by M. Mantoiu and M. Ruzhansky, a global pseudo-differential calculus has been developed for unimodular groups of type I. In the present article we generalize the main results to arbitrary locally compact groups of type I. Our methods involve the use of Plancherel's theorem for non-unimodular groups. We also make connections with a $C^*$-algebraic formalism, involving dynamical systems, and give explicit constructions for the group of affine transformations of the real line.