Quantum Harmonic Analysis on locally compact abelian groups

TL;DR

This paper generalizes quantum harmonic analysis to locally compact abelian groups with Heisenberg multipliers, including non-second countable spaces, and extends key theorems like Wiener's approximation to operators on coorbit spaces.

Contribution

It extends quantum harmonic analysis to broader abelian phase spaces, including non-second countable cases, and generalizes Wiener's approximation theorem for operators on coorbit spaces.

Findings

Extended Werner's quantum harmonic analysis to non-second countable phase spaces.

Generalized Wiener's approximation theorem to operators on coorbit spaces.

Achieved a unified harmonic analysis framework for functions and operators on abelian groups.

Abstract

We extend the notions of quantum harmonic analysis, as introduced in R. Werner's paper from 1984 (J. Math. Phys. 25(5)), to abelian phase spaces, by which we mean a locally compact abelian group endowed with a Heisenberg multiplier. In this way, we obtain a joint harmonic analysis of functions and operators for each such phase space. For all this, we spend significant extra effort to include also phase spaces which are not second countable. We obtain most results from Werner's paper for these general phase spaces, up to Wiener's approximation theorem for operators. As an addition, we extend certain of those results (most notably Wiener's approximation theorem) to operators acting on certain coorbit spaces affiliated with the phase space.