Quantum Harmonic Analysis on locally compact abelian groups

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.