Spectral distances: Results for Moyal plane and noncommutative torus
Résumé
The spectral distance for noncommutative Moyal planes in the framework of a non compact spectral triple recently proposed as a possible noncommutaitve analog of non compact Riemannian spin manifold is considered. An explicit formula for the distance between any two elements of a particular class of pure states can be determined. The corresponding result is discussed and somewhat interpreted. The existence of some pure states at infinite distance signals that the topology of the spectral distance on the space of states is not the weak * topology. The case of the noncommutative torus, both in the irrational and rational case, is also considered and some determination of the spectral distance between some pure and non-pure states is also given.