In this talk I discuss a link between the Moyal plane and anti-Wick operators, a special class of pseudodifferential operators.

The Moyal plane is viewed as a noncommutative manifold, concretely as a group C*-algebra, equipped with canonical derivations and projective modules with canonical derivations. The presentation will link smooth subalgebras on the Moyal plane with well-known classes of function spaces, the modulation spaces of Feichtinger. In this setting anti-Wick operators enter as differential operators on vector bundles over the Moyal plane.