  Quantum harmonic analysis on phase space is shown to be linked with
localization operators. The convolution between operators and the convolution
between a function and an operator provide a conceptual framework for the
theory of localization operators which is complemented by an appropriate
Fourier transform, the Fourier-Wigner transform. We use Lieb's uncertainty
principle to establish a sharp Hausdorff-Young inequality for the
Fourier-Wigner transform. Noncommutative Tauberian theorems due to Werner allow
us to extend results of Bayer and Gr\"ochenig on localization operators.
Furthermore we show that the Arveson spectrum and the theory of Banach modules
provide the abstract setting of quantum harmonic analysis.
