Curved Noncommutative Tori as Leibniz Quantum Compact Metric Spaces

We prove that curved noncommutative tori, introduced by Dabrowski and Sitarz, are Leibniz quantum compact metric spaces and that they form a continuous family over the group of invertible matrices with entries in the commutant of the quantum tori in the regular representation, when this group is endowed with a natural length function.Comment: 16 Pages, v3: accepted in Journal of Math. Physic

Convergence of fuzzy tori and quantum tori for the quantum Gromov–Hausdorff propinquity: An explicit approach

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Connes distance by examples: Homothetic spectral metric spaces

We study metric properties stemming from the Connes spectral distance on three types of non compact noncommutative spaces which have received attention recently from various viewpoints in the physics literature. These are the noncommutative Moyal plane, a family of harmonic Moyal spectral triples for which the Dirac operator squares to the harmonic oscillator Hamiltonian and a family of spectral triples with Dirac operator related to the Landau operator. We show that these triples are homothetic spectral metric spaces, having an infinite number of distinct pathwise connected components. The homothetic factors linking the distances are related to determinants of effective Clifford metrics. We obtain as a by product new examples of explicit spectral distance formulas. The results are discussed.Comment: 23 pages. Misprints corrected, references updated, one remark added at the end of the section 3. To appear in Review in Mathematical Physic