Abstract
We construct a class of compact quantum metric spaces. We prove that twisted reduced group C*-algebras for discrete groups with twisted rapid decay property are compact quantum metric spaces, which contain noncommutative tori, hyperbolic reduced group C*-algebras and discrete Heisenberg group C*-algebras, and that the compact quantum metric space structures depend only on the cohomology class of 2-cocycles in the Lipschitz isometric sense.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Antonescu C., Christensen E.: Metrics on group C*-algebras and a non-commutative Arzelà-Ascoli theorem. J. Funct. Anal. 214(2), 247–259 (2004)
Chatterji I., Ruane K.: Some geometric groups with rapid decay. Geom. Funct. Anal. 15(2), 311–339 (2005)
Christ, M., Rieffel, M.A.: Nilpotent group C*-algebras as compact quantum metric spaces. arXiv:1508.00980
Connes A.: Compact metric spaces, Fredholm modules, and hyperfiniteness. Ergodic Theory Dyn. Syst. 9(2), 207–220 (1989)
Connes, A.: Noncommutative Geometry. Academic Press, Inc., San Diego, CA, xiv+664 pp (1994)
Elliott G.A., Li H.: Strong Morita equivalence of higher-dimensional noncommutative tori. II. Math. Ann. 341(4), 825–844 (2008)
Haagerup U.: An example of a non-nuclear C*-algebra which has the metric approximation property. Invent. Math. 50(3), 279–293 (1979)
de la Harpe P.: Groupes hyperboliques, algèbres d’opérateurs et un théorème de Jolissaint. C. R. Acad. Sci. Paris Sér. I Math. 307(14), 771–774 (1988)
Jolissaint P.: Rapidly decreasing functions in reduced C*-algebras of groups. Trans. Am. Math. Soc. 317(1), 167–196 (1990)
Kerr D.: Dimension and dynamical entropy for metrized C*-algebras. Commun. Math. Phys. 232(3), 501–534 (2003)
Lee S.T., Packer J.A.: The cohomology of the integer Heisenberg groups. J. Algebra 184(1), 230–250 (1996)
Mathai V.: Heat kernels and the range of the trace on completions of twisted group algebras, with an appendix by Indira Chatterji. Contemp. Math. 398, 321–345 (2006)
Ozawa N., Rieffel M.A.: Hyperbolic group C*-algebras and free-product C*-algebras as compact quantum metric spaces. Can. J. Math. 57(5), 1056–1079 (2005)
Pavlović B.: Defining metric spaces via operators from unital C*-algebras. Pac. J. Math. 186(2), 285–313 (1998)
Rieffel, M.A.: Non-commutative tori—a case study of non-commutative differentiable manifolds, In: Geometric and Topological Invariants of Elliptic Operators (Brunswick, ME, 1988). Contemp. Math., vol. 105, pp. 191–211. Amer. Math. Soc., Providence, RI (1990)
Rieffel M.A.: Metrics on states from actions of compact groups. Doc. Math. 3, 215–229 (1998)
Rieffel M.A.: Metrics on state spaces. Doc. Math. 4, 559–600 (1999)
Rieffel M.A.: Group C*-algebras as compact quantum metric spaces. Doc. Math. 7, 605–651 (2002)
Rieffel M.A.: Gromov–Hausdorff distance for quantum metric spaces. Mem. Am. Math. Soc. 168(796), 1–65 (2004)
Wolf J.A.: Growth of finitely generated solvable groups and curvature of Riemannian manifold. J. Differ. Geom. 2, 421–446 (1968)
Wu W.: Lipschitzness of *-homomorphisms between C*-metric algebras. Sci. China Math. 54(11), 2473–2485 (2011)
Zeller-Meier G.: Produits croisés d’une C*-algèbre par un groupe d’automorphismes. J. Math. Pures Appl. 9(47), 101–239 (1968)
Author information
Authors and Affiliations
Corresponding author
Additional information
The research was supported by National Natural Science Foundation of China (Grant No. 11171109), and by Science and Technology Commission of Shanghai Municipality (Grant No. 13dz2260400).
Rights and permissions
About this article
Cite this article
Long, B., Wu, W. Twisted Group C*-Algebras as Compact Quantum Metric Spaces. Results Math 71, 911–931 (2017). https://doi.org/10.1007/s00025-016-0562-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00025-016-0562-7