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.

我们演示了如何使用时频分析中出现的一类权重，为第二可数局部紧致阿贝尔群的相空间中的扭曲组\（C ^ * \） -格的代数构造谱三元组。这产生了一种在Heisenberg模上构造量子\（C ^ k \）-结构的方法，并且我们展示了如何使用Gabor分析和Feichtinger代数的某些加权类似物来获得这种结构。我们将非交换2-tori的标准光谱三重态作为一种特殊情况，作为另一个示例，我们在非交换螺线管上定义了一个光谱三重态，并在相关的Heisenberg模块上定义了一个量子\（C ^ k \）-结构。