Abstract
We construct spectral triples on C*-algebraic extensions of unital C*-algebras by stable ideals satisfying a certain Toeplitz type property using given spectral triples on the quotient and ideal. Our construction behaves well with respect to summability and produces new spectral quantum metric spaces out of given ones. Using our construction we find new spectral triples on the quantum 2- and 3-spheres giving a new perspective on these algebras in noncommutative geometry.
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Baaj S., Julg P.: Théorie bivariante de Kasparov et opérateurs non bornés dans les \({C^{\ast}}\)-modules hilbertiens. C. R. Acad. Sci. Paris Sér. I Math. 296(21), 875–878 (1983)
Bellissard, J.V., Marcolli, M., Reihani, K.: Dynamical systems on spectral metric spaces. arXiv:1008.4617 [math.OA] (2010)
Blackadar, B.: K-Theory for Operator Algebras, Volume 5 of Mathematical Sciences Research Institute Publications, 2nd edn. Cambridge University Press, Cambridge (1998)
Chakraborty P.: From \({C^\ast}\)-algebra extensions to compact quantum metric spaces, quantum SU(2), Podleś spheres and other examples. J. Aust. Math. Soc. 90(1), 1–8 (2011)
Chakraborty P., Pal A.: Equivariant spectral triples on the quantum SU(2) group. K-Theory 28(2), 107–126 (2003)
Chakraborty P., Pal A.: On equivariant Dirac operators for \({{\rm SU}_q(2)}\). Proc. Indian Acad. Sci. Math. Sci. 116(4), 531–541 (2006)
Chakraborty P., Pal A.: Torus equivariant spectral triples for odd-dimensional quantum spheres coming from \({C^*}\)-extensions. Lett. Math. Phys. 80(1), 57–68 (2007)
Chakraborty P., Pal A.: Characterization of \({{\rm SU}_q(\l+1)}\)-equivariant spectral triples for the odd dimensional quantum spheres. J. Reine Angew. Math. 623, 25–42 (2008)
Christensen E.: On weakly D-differentiable operators. Expo. Math. 34(1), 27–42 (2016)
Christensen E., Ivan C.: Spectral triples for AF \({C^*}\)-algebras and metrics on the Cantor set. J. Oper. Theory 56(1), 17–46 (2006)
Christensen E., Ivan C.: Extensions and degenerations of spectral triples. Commun. Math. Phys. 285(3), 925–955 (2009)
Connes A.: Compact metric spaces, Fredholm modules, and hyperfiniteness. Ergodic Theory Dynam. Syst. 9(2), 207–220 (1989)
Connes A.: Noncommutative Geometry. Academic Press Inc., San Diego (1994)
Connes A.: Cyclic cohomology, quantum group symmetries and the local index formula for \({SU_q(2)}\). J. Inst. Math. Jussieu 3, 17–68 (2004)
Connes A.: On the spectral characterization of manifolds. J. Noncommut. Geom. 7(1), 1–82 (2013)
Connes A., Moscovici H.: The local index formula in noncommutative geometry. Geom. Funct. Anal. 5(2), 174–243 (1995)
Dabrowski L., D’Andrea F., Landi G., Wagner E.: Dirac operators on all Podleś quantum spheres. J. Noncommut. Geom. 1(2), 213–239 (2007)
Dabrowski L., Landi G., Sitarz A., Suijlekom W., Varilly J.C.: The Dirac operator on \({SU_q(2)}\). Commun. Math. Phys. 259, 729–759 (2005)
Gabriel, O., Grensing, M.: Spectral triples and generalized crossed products. arXiv:1310.5993 [math.OA] (2013)
Goffeng, M., Mesland, B.: Spectral triples and finite summability on Cuntz-Krieger algebras. Doc. Math 20, 89–170 (2015) (electronic)
Hawkins A., Skalski A., White S., Zacharias J.: On spectral triples on crossed products arising from equicontinuous actions. Math. Scand. 113(2), 262–291 (2013)
Higson N., Roe J.: Analytic K-Homology. Oxford Mathematical Monographs. Oxford University Press, Oxford (2000)
Kasparov G.: The operator K-functor and extensions of \({C^{\ast}}\)-algebras. Izv. Akad. Nauk SSSR Ser. Mat. 44(3), 571–636, 719 (1980)
Latrémolière F.: Bounded-Lipschitz distances on the state space of a \({C^*}\)-algebra. Taiwan. J. Math. 11(2), 447–469 (2007)
Latrémolière F.: Quantum locally compact metric spaces. J. Funct. Anal. 264(1), 362–402 (2013)
Lord S., Rennie A., Várilly J. C.: Riemannian manifolds in noncommutative geometry. J. Geom. Phys. 62(7), 1611–1638 (2012)
Neshveyev S., Tuset L.: The Dirac operator on compact quantum groups. J. Reine Angew. Math. 641, 1–20 (2010)
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)
Podleś P.: Quantum spheres. Lett. Math. Phys. 14(3), 193–202 (1987)
Rennie A.: Summability for nonunital spectral triples. K-Theory 31(1), 71–100 (2004)
Rennie, A.: Spectral triples: examples and applications, notes for lectures given at the workshop on noncommutative geometry and physics, Yokohama (2009)
Rennie, A., Varilly, J.C.: Reconstruction of manifolds in noncommutative geometry. arXiv:math/0610418 [math.OA] (2006)
Rieffel, M.A.: Metrics on states from actions of compact groups. Doc. Math. 3, 215–229 (1998) (electronic)
Rieffel, M.A.: Metrics on state spaces. Doc. Math. 4, 559–600 (1999) (electronic)
Rieffel, M.A.: Compact quantum metric spaces. In: Operator Algebras, Quantization, and Noncommutative Geometry, Volume 365 of Contemp. Math., pp. 315–330. Amer. Math. Soc., Providence (2004)
Rieffel, M.A.: Gromov–Hausdorff distance for quantum metric spaces. In: Matrix Algebras Converge to the Sphere for Quantum Gromov–Hausdorff Distance. American Mathematical Society, Providence 2004. Mem. Amer. Math. Soc. 168, no. 796 (2004)
Roytenberg D.: Poisson cohomology of \({SU(2)}\)-covariant “necklace” poisson structures on \({S^2}\). J. Nonlinear Math. Phys. 9(3), 347–356 (2002)
van Suijlekom W., Dabrowski L., Landi G., Sitarz A., Varilly J.C.: The local index formula for \({SU_q(2)}\). K-Theory 35, 375–394 (2005)
Várilly, J.C., Witkowski, P.: Dirac operators and spectral geometry (2006)
Wang X.: Voiculescu theorem, Sobolev lemma, and extensions of smooth algebras. Bull. Am. Math. Soc. (N.S.) 27(2), 292–297 (1992)
Woronowicz S.L.: Twisted \({{\rm SU}(2)}\) group. An example of a noncommutative differential calculus. Publ. Res. Inst. Math. Sci. 23(1), 117–181 (1987)
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Communicated by Y. Kawahigashi
Supported by: EPSRC Grant EP/I019227/1-2.
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Hawkins, A., Zacharias, J. Spectral Metric Spaces on Extensions of C*-Algebras. Commun. Math. Phys. 350, 475–506 (2017). https://doi.org/10.1007/s00220-016-2820-7
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DOI: https://doi.org/10.1007/s00220-016-2820-7