Quantum metric spaces of quantum maps

Yıl 2018, Cilt: 1 Sayı: 1, 54 - 60, 11.03.2018

Maysam Maysami Sadr

https://doi.org/10.32323/ujma.382008

Cited By: 2

Öz

We show that any quantum family of quantum maps from a noncommutative space to a compact quantum metric space has a canonical quantum pseudo-metric structure. Here by a 'compact quantum metric space' we mean a unital C*-algebra together with a Lipschitz seminorm, in the sense of Rieffel, which induces the weak* topology on the state space of the C*-algebra. Our main result generalizes a classical result to noncommutative world.

Anahtar Kelimeler

Compact quantum metric space, C*-algebra, Lipschitz algebra, Quantum family of maps, State space

Kaynakça

• [1] A. Connes, Compact metric spaces, Fredholm modules and hyperfiniteness, Ergo. Th. Dyn. Sys. 9 (1989), 207–220.
• [2] A. Connes, Noncommutative Geometry, Academic Press, 1994.
• [3] G. Kuperberg, N. Weaver, A von Neumann algebra approach to quantum metrics/quantum relations, Vol. 215, no. 1010. American Mathematical Society, 2012.
• [4] S. T. Rachev, Probability Metrics and the Stability of Stochastic Models, John Wiley and Sons, 1991.
• [5] M. A. Rieffel, Metrics on states from actions of compact groups, Doc. Math. 3 (1998), 215–229.
• [6] M. A. Rieffel, Metrics on state spaces, Doc. Math. 4 (1999), 559–600.
• [7] M. A. Rieffel, Gromov-Hausdorff distance for quantum metric spaces, Mem. Amer. Math. Soc. 168 (2004), 1–65.
• [8] M. A. Rieffel, Compact quantum metric spaces, Contemp. Math. 365 (2004), 315–330.
• [9] M. A. Rieffel, Leibniz seminorms for Matrix algebras converge to the sphere, Quanta of maths, Clay Math. Proc., vol. 11, Amer. Math. Soc., Providence, RI, 2010, 543–578.
• [10] M. M. Sadr, Quantum functor Mor, Math. Pannonica 21 no. 1 (2010), 77–88.
• [11] M. M. Sadr, A kind of compact quantum semigroups, Int. J. Math. Math. Sci. 2012 (2012), Article ID 725270, 10 pages.
• [12] M. M. Sadr, On the quantum groups and semigroups of maps between noncommutative spaces, Czechoslovak Math. J. 67 no. 1 (2017), 97–121.
• [13] M. M. Sadr, Quantum metrics on noncommutative spaces, available at https://arxiv.org/pdf/1606.00661.pdf
• [14] M. M. Sadr, Metric operator fields, available at https://arxiv.org/pdf/1705.03378.pdf
• [15] P. M. Sołtan, Quantum families of maps and quantum semigroups on finite quantum spaces, J. Geom. Phys. 59 (2009), 354–368.
• [16] S. L. Woronowicz, Pseudogroups, pseudospaces and Pontryagin duality, Proceedings of the International Conference on Mathematical Physics, Lausanne 1979 , Lecture Notes in Physics 116, 407–412.