Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 6 (2010), 026, 17 pages      arXiv:0912.4185      http://dx.doi.org/10.3842/SIGMA.2010.026
Contribution to the Proceedings of the XVIIIth International Colloquium on Integrable Systems and Quantum Symmetries

Spectral Distances: Results for Moyal Plane and Noncommutative Torus

Eric Cagnache and Jean-Christophe Wallet
Laboratoire de Physique Théorique, Bât. 210, CNRS, Université Paris-Sud 11, F-91405 Orsay Cedex, France

Received October 31, 2009, in final form March 20, 2010; Published online March 24, 2010

Abstract
The spectral distance for noncommutative Moyal planes is considered in the framework of a non compact spectral triple recently proposed as a possible noncommutative analog of non compact Riemannian spin manifold. 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. 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 is also considered and a formula for the spectral distance between some states is also obtained.

Key words: noncommutative geometry; non-compact spectral triples; spectral distance; noncommutative torus; Moyal planes.

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