Band-dominated and Fourier-band-dominated operators on locally compact abelian groups

By relating notions from quantum harmonic analysis and band-dominated operator theory, we prove that over any locally compact abelian group $G$, the operator algebra $\mathcal C_1$ from quantum harmonic analysis agrees with the intersection of band-dominated operators and Fourier band-dominated operators. As an application, we characterize the compactness of operators acting on $L^2(G)$ and compare it with previous results in the discrete case. In particular, our results can be seen as a generalization of the limit operator concept to the non-discrete world. Moreover, we briefly discuss property $A'$ for arbitrary locally compact abelian groups.