Abstract
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.
Highlights
The interplay between Gabor analysis and noncommutative geometry [8] has been explored earlier and has recently attracted some interest, see for example [2,3,4, 10, 11, 22, 23, 25, 28, 29]
Results in Gabor analysis supply interesting examples of structures studied in operator algebra theory and noncommutative geometry
For the noncommutative 2-torus, we show that our approach yields an equal QCk-structure as if using the standard spectral triple, and for the noncommutative solenoid, our construction provides a definition of smoothness which so far has not appeared in the literature
Summary
The interplay between Gabor analysis and noncommutative geometry [8] has been explored earlier and has recently attracted some interest, see for example [2,3,4, 10, 11, 22, 23, 25, 28, 29]. Problems in Gabor analysis can often effectively be rephrased as operator algebraic questions. Gabor analysis provides a way to generate projective modules over noncommutative tori [28]. Results in Gabor analysis supply interesting examples of structures studied in operator algebra theory and noncommutative geometry. The main part of this paper focuses on the latter aspect
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