Modulation Spaces as a Smooth Structure in Noncommutative Geometry

We demonstrate how to construct spectral triples for twisted group $$C^*$$ -algebras of lattices in phase space of a second-countable locally compact abelian group using a class of weights appearing in time–frequency analysis. This yields a way of constructing quantum $$C^k$$ -structures on Heisenberg modules, and we show how to obtain such structures using Gabor analysis and certain weighted analogues of Feichtinger’s algebra. We treat the standard spectral triple for noncommutative 2-tori as a special case, and as another example we define a spectral triple on noncommutative solenoids and a quantum $$C^k$$ -structure on the associated Heisenberg modules.