Transversals, duality, and irrational rotation
HTML articles powered by AMS MathViewer
- by Anna Duwenig and Heath Emerson HTML | PDF
- Trans. Amer. Math. Soc. Ser. B 7 (2020), 254-289
Abstract:
An early result of Noncommutative Geometry was Connes’ observation in the 1980’s that the Dirac-Dolbeault cycle for the $2$-torus $\mathbb {T}^2$, which induces a Poincaré self-duality for $\mathbb {T}^2$, can be ‘quantized’ to give a spectral triple and a K-homology class in $\mathrm {KK}_0(A_\theta \otimes A_\theta , \mathbb {C})$ providing the co-unit for a Poincaré self-duality for the irrational rotation algebra $A_\theta$ for any $\theta \in \mathbb {R}\setminus \mathbb {Q}$. Connes’ proof, however, relied on a K-theory computation and does not supply a representative cycle for the unit of this duality. Since such representatives are vital in applications of duality, we supply such a cycle in unbounded form in this article. Our approach is to construct, for any non-zero integer $b$, a finitely generated projective module $\mathcal {L}_{b}$ over $A_\theta \otimes A_\theta$ by using a reduction-to-a-transversal argument of Muhly, Renault, and Williams, applied to a pair of Kronecker foliations along lines of slope $\theta$ and $\theta + b$, using the fact that these flows are transverse to each other. We then compute Connes’ dual of $[\mathcal {L}_{b}]$ and prove that we obtain an invertible $\tau _{b}\in \mathrm {KK}_0(A_\theta , A_\theta )$, represented by an equivariant bundle of Dirac-Schrödinger operators. An application of equivariant Bott Periodicity gives a form of higher index theorem describing functoriality of such ‘$b$-twists’ and this allows us to describe the unit of Connes’ duality in terms of a combination of two constructions in KK-theory. This results in an explicit spectral representative of the unit – a kind of ‘quantized Thom class’ for the diagonal embedding of the noncommutative torus.References
- Berndt A. Brenken, Representations and automorphisms of the irrational rotation algebra, Pacific J. Math. 111 (1984), no. 2, 257–282. MR 734854, DOI 10.2140/pjm.1984.111.257
- Alain Connes, Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994. MR 1303779
- Alain Connes, Gravity coupled with matter and the foundation of non-commutative geometry, Comm. Math. Phys. 182 (1996), no. 1, 155–176. MR 1441908, DOI 10.1007/BF02506388
- A. Connes and G. Skandalis, The longitudinal index theorem for foliations, Publ. Res. Inst. Math. Sci. 20 (1984), no. 6, 1139–1183. MR 775126, DOI 10.2977/prims/1195180375
- A. Duwenig, Poincaré self-duality of $A_{\theta }$, 2020, URI: http://hdl.handle.net/1828/11678
- Siegfried Echterhoff, Heath Emerson, and Hyun Jeong Kim, $KK$-theoretic duality for proper twisted actions, Math. Ann. 340 (2008), no. 4, 839–873. MR 2372740, DOI 10.1007/s00208-007-0171-6
- Heath Emerson, Lefschetz numbers for $C^*$-algebras, Canad. Math. Bull. 54 (2011), no. 1, 82–99. MR 2797970, DOI 10.4153/CMB-2010-084-5
- Heath Emerson, Noncommutative Poincaré duality for boundary actions of hyperbolic groups, J. Reine Angew. Math. 564 (2003), 1–33. MR 2021032, DOI 10.1515/crll.2003.090
- Heath Emerson, The class of a fibre in noncommutative geometry, J. Geom. Phys. 148 (2020), 103537, 37. MR 4043062, DOI 10.1016/j.geomphys.2019.103537
- José M. Gracia-Bondía, Joseph C. Várilly, and Héctor Figueroa, Elements of noncommutative geometry, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Boston, Inc., Boston, MA, 2001. MR 1789831, DOI 10.1007/978-1-4612-0005-5
- Jens Kaad, On the unbounded picture of $KK$-theory, SIGMA Symmetry Integrability Geom. Methods Appl. 16 (2020), Paper No. 082, 21. MR 4137615, DOI 10.3842/SIGMA.2020.082
- Jerome Kaminker and Ian Putnam, $K$-theoretic duality of shifts of finite type, Comm. Math. Phys. 187 (1997), no. 3, 509–522. MR 1468312, DOI 10.1007/s002200050147
- Jerome Kaminker, Ian F. Putnam, and Michael F. Whittaker, K-theoretic duality for hyperbolic dynamical systems, J. Reine Angew. Math. 730 (2017), 263–299. MR 3692021, DOI 10.1515/crelle-2014-0126
- G. G. Kasparov, Equivariant $KK$-theory and the Novikov conjecture, Invent. Math. 91 (1988), no. 1, 147–201. MR 918241, DOI 10.1007/BF01404917
- Dan Kucerovsky, The $KK$-product of unbounded modules, $K$-Theory 11 (1997), no. 1, 17–34. MR 1435704, DOI 10.1023/A:1007751017966
- Matthias Lesch and Bram Mesland, Sums of regular self-adjoint operators in Hilbert-$C^*$-modules, J. Math. Anal. Appl. 472 (2019), no. 1, 947–980. MR 3906406, DOI 10.1016/j.jmaa.2018.11.059
- Wolfgang Lück and Jonathan Rosenberg, Equivariant Euler characteristics and $K$-homology Euler classes for proper cocompact $G$-manifolds, Geom. Topol. 7 (2003), 569–613. MR 2026542, DOI 10.2140/gt.2003.7.569
- Bram Mesland, Unbounded bivariant $K$-theory and correspondences in noncommutative geometry, J. Reine Angew. Math. 691 (2014), 101–172. MR 3213549, DOI 10.1515/crelle-2012-0076
- Ralf Meyer and Ryszard Nest, The Baum-Connes conjecture via localisation of categories, Topology 45 (2006), no. 2, 209–259. MR 2193334, DOI 10.1016/j.top.2005.07.001
- Paul S. Muhly, Jean N. Renault, and Dana P. Williams, Equivalence and isomorphism for groupoid $C^\ast$-algebras, J. Operator Theory 17 (1987), no. 1, 3–22. MR 873460
- Ian F. Putnam and Jack Spielberg, The structure of $C^\ast$-algebras associated with hyperbolic dynamical systems, J. Funct. Anal. 163 (1999), no. 2, 279–299. MR 1680475, DOI 10.1006/jfan.1998.3379
- Graham A. Niblo, Roger Plymen, and Nick Wright, Poincaré duality and Langlands duality for extended affine Weyl groups, Ann. K-Theory 3 (2018), no. 3, 491–522. MR 3830200, DOI 10.2140/akt.2018.3.491
- Jean Renault, A groupoid approach to $C^{\ast }$-algebras, Lecture Notes in Mathematics, vol. 793, Springer, Berlin, 1980. MR 584266, DOI 10.1007/BFb0091072
- Lachlan MacDonald and Adam Rennie, The Godbillon-Vey invariant and equivariant $KK$-theory, Ann. K-Theory 5 (2020), no. 2, 249–294. MR 4113770, DOI 10.2140/akt.2020.5.249
- Adam Rennie, David Robertson, and Aidan Sims, Poincaré duality for Cuntz-Pimsner algebras, Adv. Math. 347 (2019), 1112–1172. MR 3924388, DOI 10.1016/j.aim.2019.02.032
- John Roe, Elliptic operators, topology and asymptotic methods, 2nd ed., Pitman Research Notes in Mathematics Series, vol. 395, Longman, Harlow, 1998. MR 1670907
- Yasuo Watatani, Toral automorphisms on irrational rotation algebras, Math. Japon. 26 (1981), no. 4, 479–484. MR 634924
Additional Information
- Anna Duwenig
- Affiliation: School of Mathematics and Applied Statistics, University of Wollongong, Northfields Ave, Wollongong, NSW 2522, Australia
- MR Author ID: 1378284
- ORCID: 0000-0001-6042-2561
- Email: aduwenig@uow.edu.au
- Heath Emerson
- Affiliation: Department of Mathematics and Statistics, University of Victoria, P.O. Box 3045 STN CSC, Victoria, British Columbia, V8W 3P4 Canada
- MR Author ID: 630788
- Email: hemerson@math.uvic.ca
- Received by editor(s): March 20, 2020
- Received by editor(s) in revised form: July 16, 2020
- Published electronically: December 2, 2020
- Additional Notes: This research was supported by an NSERC Discovery grant.
- © Copyright 2020 by the authors under Creative Commons Attribution 3.0 License (CC BY 3.0)
- Journal: Trans. Amer. Math. Soc. Ser. B 7 (2020), 254-289
- MSC (2020): Primary 19K35; Secondary 58B34, 46L08
- DOI: https://doi.org/10.1090/btran/54
- MathSciNet review: 4181521