The Cuntz-Pimsner extension and mapping cone exact sequences
DOI:
https://doi.org/10.7146/math.scand.a-115634Abstract
For Cuntz-Pimsner algebras of bi-Hilbertian bimodules with finite Jones-Watatani index satisfying some side conditions, we give an explicit isomorphism between the $K$-theory exact sequences of the mapping cone of the inclusion of the coefficient algebra into a Cuntz-Pimsner algebra, and the Cuntz-Pimsner exact sequence. In the process we extend some results by the second author and collaborators from finite projective bimodules to certain finite index bimodules, and also clarify some aspects of Pimsner's `extension of scalars' construction.
References
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Arici, F., Brain, S., and Landi, G., The Gysin sequence for quantum lens spaces, J. Noncommut. Geom. 9 (2015), no. 4, 1077–1111. https://doi.org/10.4171/JNCG/216
Arici, F., Kaad, J., and Landi, G., Pimsner algebras and Gysin sequences from principal circle actions, J. Noncommut. Geom. 10 (2016), no. 1, 29–64. https://doi.org/10.4171/JNCG/228
Baaj, S. and 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 (1983), no. 21, 875–878.
Blackadar, B., $K$-theory for operator algebras, second ed., Mathematical Sciences Research Institute Publications, vol. 5, Cambridge University Press, Cambridge, 1998.
Blackadar, B., Operator algebras: Theory of $C^*$-algebras and von Neumann algebras, operator algebras and non-commutative geometry, III, Encyclopaedia of Mathematical Sciences, vol. 122, Springer-Verlag, Berlin, 2006. https://doi.org/10.1007/3-540-28517-2
Brown, L. G., Mingo, J. A., and Shen, N.-T., Quasi-multipliers and embeddings of Hilbert $C^\ast $-bimodules, Canad. J. Math. 46 (1994), no. 6, 1150–1174. https://doi.org/10.4153/CJM-1994-065-5
Carey, A. L., Neshveyev, S., Nest, R., and Rennie, A., Twisted cyclic theory, equivariant $KK$-theory and KMS states, J. Reine Angew. Math. 650 (2011), 161–191. https://doi.org/10.1515/CRELLE.2011.007
Carey, A. L., Phillips, J., and Rennie, A., Noncommutative Atiyah-Patodi-Singer boundary conditions and index pairings in $KK$-theory, J. Reine Angew. Math. 643 (2010), 59–109. https://doi.org/10.1515/CRELLE.2010.045
Cuntz, J. and Skandalis, G., Mapping cones and exact sequences in $KK$-theory, J. Operator Theory 15 (1986), no. 1, 163–180.
Drinen, D. and Tomforde, M., Computing $K$-theory and $\rm Ext$ for graph $C^*$-algebras, Illinois J. Math. 46 (2002), no. 1, 81–91.
Goffeng, M., Mesland, B., and Rennie, A., Shift-tail equivalence and an unbounded representative of the Cuntz-Pimsner extension, Ergodic Theory Dynam. Systems 38 (2018), no. 4, 1389–1421. https://doi.org/10.1017/etds.2016.75
Kajiwara, T., Pinzari, C., and Watatani, Y., Jones index theory for Hilbert $C^*$-bimodules and its equivalence with conjugation theory, J. Funct. Anal. 215 (2004), no. 1, 1–49. https://doi.org/10.1016/j.jfa.2003.09.008
Kasparov, G. G., The operator $K$-functor and extensions of $C^\ast $-algebras, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 3, 571–636.
Katsura, T., A construction of $C^*$-algebras from $C^*$-correspondences, in “Advances in quantum dynamics (South Hadley, MA, 2002)”, Contemp. Math., vol. 335, Amer. Math. Soc., Providence, RI, 2003, pp. 173–182. https://doi.org/10.1090/conm/335/06007
Katsura, T., On $C^*$-algebras associated with $C^*$-correspondences, J. Funct. Anal. 217 (2004), no. 2, 366–401. https://doi.org/10.1016/j.jfa.2004.03.010
Lledó, F. and Vasselli, E., On the nuclearity of certain Cuntz-Pimsner algebras, Math. Nachr. 283 (2010), no. 5, 752–757. https://doi.org/10.1002/mana.200710011
Meyer, R. and Nest, R., The Baum-Connes conjecture via localisation of categories, Topology 45 (2006), no. 2, 209–259. https://doi.org/10.1016/j.top.2005.07.001
Pimsner, M. V., A class of $C^*$-algebras generalizing both Cuntz-Krieger algebras and crossed products by $\bf Z$, in “Free probability theory (Waterloo, ON, 1995)”, Fields Inst. Commun., vol. 12, Amer. Math. Soc., Providence, RI, 1997, pp. 189–212.
Putnam, I. F., An excision theorem for the $K$-theory of $C^*$-algebras, J. Operator Theory 38 (1997), no. 1, 151–171.
Raeburn, I. and Williams, D. P., Morita equivalence and continuous-trace $C^*$-algebras, Mathematical Surveys and Monographs, vol. 60, American Mathematical Society, Providence, RI, 1998. https://doi.org/10.1090/surv/060
Rennie, A., Robertson, D., and Sims, A., The extension class and KMS states for Cuntz-Pimsner algebras of some bi-Hilbertian bimodules, J. Topol. Anal. 9 (2017), no. 2, 297–327. https://doi.org/10.1142/S1793525317500108
Rennie, A., Robertson, D., and Sims, A., Poincaré duality for Cuntz-Pimsner algebras, Adv. Math. 347 (2019), 1112–1172. https://doi.org/10.1016/j.aim.2019.02.032
Rennie, A. and Sims, A., Non-commutative vector bundles for non-unital algebras, SIGMA Symmetry Integrability Geom. Methods Appl. 13 (2017), paper No. 041, 12 pp. https://doi.org/10.3842/SIGMA.2017.041
Rosenberg, J. and Schochet, C., The Künneth theorem and the universal coefficient theorem for Kasparov's generalized $K$-functor, Duke Math. J. 55 (1987), no. 2, 431–474. https://doi.org/10.1215/S0012-7094-87-05524-4
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Published
2019-10-19
How to Cite
Arici, F., & Rennie, A. (2019). The Cuntz-Pimsner extension and mapping cone exact sequences. MATHEMATICA SCANDINAVICA, 125(2), 291–319. https://doi.org/10.7146/math.scand.a-115634
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