On Modulation and Translation Invariant Operators and the Heisenberg Module

We investigate spaces of operators which are invariant under translations or modulations by lattices in phase space. The natural connection to the Heisenberg module is considered, giving results on the characterisation of such operators as limits of finite--rank operators. Discrete representations of these operators in terms of elementary objects and the composition calculus are given. Different quantisation schemes are discussed with respect to the results.