Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-28T04:54:55.780Z Has data issue: false hasContentIssue false
  • English
  • Français

Vector Bundles and Gromov–Hausdorff Distance

Published online by Cambridge University Press:  25 August 2009

Marc A. Rieffel
Marc A. Rieffel
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720-3840, U. S., [email protected]
Get access

Abstract

We show how to make precise the vague idea that for compact metric spaces that are close together for Gromov–Hausdorff distance, suitable vector bundles on one metric space will have counterpart vector bundles on the other. Our approach employs the Lipschitz constants of projection-valued functions that determine vector bundles. We develop some computational techniques, and we illustrate our ideas with simple specific examples involving vector bundles on the circle, the two-torus, the two-sphere, and finite metric spaces. Our topic is motivated by statements concerning “monopole bundles” over matrix algebras in the literature of theoretical high-energy physics.

Keywords

vector bundlesGromov–Hausdorff distanceLipschitzprojectionsmonopole bundle
Type
Research Article
Information
Journal of K-Theory , Volume 5 , Issue 1 , February 2010 , pp. 39 - 103
Copyright
Copyright © ISOPP 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Atiyah, M. F., K-theory, second ed., Addison-Wesley Pub., Redwood City, CA, 1989. MR 1043170 (90m: 18011)Google Scholar
2.Baez, S., Balachandran, A. P., Vaidya, S., and Ydri, B., Monopoles and solitons in fuzzy physics, Comm. Math. Phys. 208 (2000), no. 3, 787–798, arXiv:hep-th/9811169. MR 1736336 (2001f:58015)CrossRefGoogle Scholar
3.Balachandran, A. P. and Immirzi, Giorgio, Duality in fuzzy sigma models, Internat. J. Modern Phys. A 19 (2004), no. 30, 5237–5245, arXiv:hep-th/0408111. MR 2108640 (2005g:81156)CrossRefGoogle Scholar
4.Balachandran, A. P., Immirzi, Giorgio, Lee, Joohan, and Prešnajder, Peter, Dirac operators on coset spaces, J. Math. Phys. 44 (2003), no. 10, 4713–4735, arXiv:hepth/0210297. MR 2008943 (2004i:58046)CrossRefGoogle Scholar
5.Bellaiche, A., The tangent space in sub-Riemannian geometry, Sub-Riemannian geometry, Birkhäuser, Basel, 1996, pp. 1–78. MR 98a:53108CrossRefGoogle Scholar
6.Blackadar, Bruce, K-theory for operator algebras, second ed., Mathematical Sciences Research Institute Publications, vol. 5, Cambridge University Press, Cambridge, 1998. MR 1656031 (99g:46104)Google Scholar
7.Brudnyi, Alexander and Brudnyi, Yuri, Extension of Lipschitz functions defined on metric subspaces of homogeneous type, Rev. Mat. Complut. 19 (2006), no. 2, 347–359, arXiv:math.FA/0609535. MR 2241435 (2007d:54012)CrossRefGoogle Scholar
8.Carow-Watamura, Ursula, Steinacker, Harold, and Watamura, Satoshi, Monopole bundles over fuzzy complex projective spaces, J. Geom. Phys. 54 (2005), no. 4, 373–399, arXiv:hep-th/0404130.MR 2144709CrossRefGoogle Scholar
9.Cheeger, Jeff and Ebin, David G., Comparison theorems in Riemannian geometry, North-Holland Publishing Co., Amsterdam, 1975. MR 0458335 (56 #16538)Google Scholar
10.Connes, Alain, C* algèbres et géométrie différentielle, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), no. 13, A599–A604. MR 572645 (81c:46053)Google Scholar
11.Cuntz, Joachim, Meyer, Ralf and Rosenberg, Jonathan M., Topological and bivariant K-theory, Oberwolfach Seminars 36, Birkhäuser Verlag, Basel, 2007. MR 2340673Google Scholar
12.Davidson, Kenneth R., C*-algebras by example, Fields Institute Monographs, vol. 6, American Mathematical Society, Providence, RI, 1996. MR 1402012 (97i:46095)Google Scholar
13.Fell, J. M. G. and Doran, R. S., Representations of* -algebras, locally compact groups, and Banach* -algebraic bundles. Vol. 1, Academic Press Inc., Boston, MA, 1988. MR 90c:46001Google Scholar
14.Frank, Michael and Larson, David R., A module frame concept for Hilbert C*-modules, The functional and harmonic analysis of wavelets and frames (San Antonio, TX, 1999), Contemp. Math., vol. 247, Amer. Math. Soc., Providence, RI, 1999, pp. 207–233. MR 1738091 (2001b:46094)Google Scholar
15.Frank, Michael and Larson, David R., Frames in Hilbert C*-modules and C*-algebras, J. Operator Theory 48 (2002), no. 2, 273–314. MR 1938798 (2003i:42040)Google Scholar
16.Friedrich, Thomas, Dirac operators in Riemannian geometry, Graduate Studies in Mathematics, vol. 25, American Mathematical Society, Providence, RI, 2000, Translated from the 1997 German original by Andreas Nestke. MR 1777332 (2001c:58017)Google Scholar
17.Goodearl, K. R., Notes on real and complex C*-algebras, Shiva Publishing Ltd., Nantwich, 1982. MR 677280 (85d:46079)Google Scholar
18.Gracia-Bondia, J. M., Vàrilly, J. C., and Figueroa, H., Elements of noncommutative geometry, Birkhäuser Boston Inc., Boston, MA, 2001. MR 1 789 831CrossRefGoogle Scholar
19.Greene, R. E. and Wu, H., C∞ approximations of convex, subharmonic, and plurisubharmonic functions, Ann. Sci. École Norm. Sup. (4) 12 (1979), no. 1, 47–84. MR 532376 (80m:53055)CrossRefGoogle Scholar
20.Gromov, M., Metric structures for Riemannian and non-Riemannian spaces, Birkhäuser Boston Inc., Boston, MA, 1999. MR 2000d:53065Google Scholar
21.Grosse, Harald, Rupp, Christian W., and Strohmaier, Alexander, Fuzzy line bundles, the Chern character and topological charges over the fuzzy sphere, J. Geom. Phys. 42 (2002), no. 1–2, 54–63, arXiv:math-ph/0105033.MR 1894075 (2003f:58015)CrossRefGoogle Scholar
22.Hawkins, Eli, Quantization of equivariant vector bundles, Comm. Math. Phys. 202 (1999), no. 3, 517–546, arXiv:math-qa/9708030.MR 1690952 (2000j:58008)CrossRefGoogle Scholar
23.Hawkins, Eli, Geometric quantization of vector bundles and the correspondence with deformation quantization, Comm. Math. Phys. 215 (2000), no. 2, 409–432, arXiv:mathqa/9808116 and 9811049. MR 1799853 (2002a:53116)CrossRefGoogle Scholar
24.Husemoller, Dale, Fibre bundles, second ed., Springer-Verlag, New York, 1975. MR 0370578 (51 #6805)Google Scholar
25.Johnson, William B., Lindenstrauss, Joram, and Schechtman, Gideon, Extensions of Lipschitz maps into Banach spaces, Israel J. Math. 54 (1986), no. 2, 129–138. MR 852474 (87k:54021)CrossRefGoogle Scholar
26.Kadison, R. V. and Ringrose, J. R., Fundamentals of the theory of operator algebras. Vol. I, American Mathematical Society, Providence, RI, 1997, Reprint of the 1983 original. MR 98f:46001aGoogle Scholar
27.Karoubi, Max, K-theory, Die Grundlehren der Mathematischen Wissenschaften, Band 226, Springer-Verlag, Berlin, 1978. MR 0488029 (58 #7605)CrossRefGoogle Scholar
28.Kato, Tosio, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473 (34 #3324)Google Scholar
29.König, Hermann and Tomczak-Jaegermann, Nicole, Norms of minimal projections, J. Funct. Anal. 119 (1994), no. 2, 253–280. MR 1261092 (94m:46024)CrossRefGoogle Scholar
30.Koszul, Jean-Louis, Homologie et cohomologie des algébres de Lie, Bull. Soc. Math. France 78, (1950). 65–127. MR 0036511,CrossRefGoogle Scholar
31.Landi, Giovanni, Deconstructing monopoles and instantons, Rev. Math. Phys. 12 (2000), no. 10, 1367–1390, arXiv:math-ph/9812004.MR 1794672 (2001m:53044)CrossRefGoogle Scholar
32.Landi, Giovanni, Projective modules of finite type and monopoles over S2, J. Geom. Phys. 37 (2001), no. 1–2, 47–62, arXiv:math-ph/9907020.MR 1806440 (2001k:58014)CrossRefGoogle Scholar
33.Landi, Giovanni and van Suijlekom, Walter, Principal fibrations from noncommutative spheres, Comm. Math. Phys. 260 (2005), no. 1, 203–225, arXiv:math.QA/0410077. MR 2175995 (2006g:58016)CrossRefGoogle Scholar
34.Lee, James R. and Naor, Assaf, Extending Lipschitz functions via random metric partitions, Invent. Math., 160 (2005), no. 1, 59–95. MR 2129708 (2006c:54013)CrossRefGoogle Scholar
35.Li, Hanfeng, Smooth approximation of Lipschitz projections, arXiv:0810.4695.Google Scholar
36.Packer, Judith A. and Rieffel, Marc A., Wavelet filter functions, the matrix completion problem, and projective modules over C(n), J. Fourier Anal. Appl. 9 (2003), no. 2, 101–116, arXiv:math.FA/0107231.MR 1964302 (2003m:42063)CrossRefGoogle Scholar
37.Packer, Judith A. and Rieffel, Marc A., Projective multi-resolution analyses for L2(ℝ2), J. Fourier Anal. Appl. 10 (2004), no. 5, 439–464, arXiv:math.FA/0308132. MR 2093911 (2005f:46133)CrossRefGoogle Scholar
38.Petersen, Peter, V, A finiteness theorem for metric spaces, J. Differential Geom. 31 (1990), no. 2, 387–395. MR 1037407 (91d:53070)CrossRefGoogle Scholar
39.Przeslawski, Krzysztof and Yost, David, Continuity properties of selectors and Michael's theorem, Michigan Math. J. 36 (1989), no. 1, 113–134, MR 989940 (90d:49010),CrossRefGoogle Scholar
40.Reitberger, Heinrich, Leopold Vietoris (1891–2002), Notices Amer. Math. Soc. 49 (2002), no. 10, 1232–1236.Google Scholar
41.Rieffel, Marc A., C*-algebras associated with irrational rotations, Pacific J. Math. 93 (1981), no. 2, 415–429. MR 623572 (83b:46087)CrossRefGoogle Scholar
42.Rieffel, Marc A., The cancellation theorem for projective modules over irrational rotation C*-algebras, Proc. London Math. Soc. (3) 47 (1983), no. 2, 285–302. MR 703981 (85g:46085)CrossRefGoogle Scholar
43.Rieffel, Marc A., Projective modules over higher-dimensional noncommutative tori, Canad. J. Math. 40 (1988), no. 2, 257–338. MR 941652 (89m:46110)CrossRefGoogle Scholar
44.Rieffel, Marc A., Metrics on states from actions of compact groups, Doc. Math. 3 (1998), 215–229, arXiv:math.OA/9807084.MR 1647515 (99k:46126)CrossRefGoogle Scholar
45.Rieffel, Marc A., Metrics on state spaces, Doc. Math. 4 (1999), 559–600, arXiv:math.OA/9906151. MR 1727499 (2001g:46154)CrossRefGoogle Scholar
46.Rieffel, Marc A., Compact quantum metric spaces, Operator algebras, quantization, and noncommutative geometry, Contemp. Math., vol. 365, Amer. Math. Soc., Providence, RI, 2004, pp. 315–330, arXiv:math.OA/0308207. MR 2106826 (2005h:46099)Google Scholar
47.Rieffel, Marc A., Gromov-Hausdorff distance for quantum metric spaces, Mem. Amer. Math. Soc. 168 (2004), no. 796, 1–65, arXiv:math.OA/0011063. MR 2055927Google Scholar
48.Rieffel, Marc A., Matrix algebras converge to the sphere for quantum Gromov-Hausdorff distance, Mem. Amer.Math. Soc. 168 (2004), no. 796, 67–91, arXiv:math.OA/0108005. MR 2055928Google Scholar
49.Rieffel, Marc A., Lipschitz extension constants equal projection constants, Contemp. Math., vol. 414, Amer. Math. Soc., Providence, RI, 2006, pp. 147–162, arXiv:math.FA/0508097. MR 2277209 (2007k:46028)Google Scholar
50.Rieffel, Marc A., A global view of equivariant vector bundles and Dirac operators on some compact homogeneous spaces, Group Representations, Ergodic Theory, and Mathematical Physics, Contemp. Math., vol. 449, Amer. Math. Soc., Providence, RI, 2008, pp. 399–415, arXiv:math.DG/0703496 .MR 2391813 (the latest arXiv version contains important corrections compared to the published version).CrossRefGoogle Scholar
51.Rieffel, Marc A., Leibniz seminorms for “Matrix algebras converge to the sphere”, arXiv:0707.3229[math.OA].Google Scholar
52.Rördam, M., Larsen, F., and Laustsen, N., An introduction to K-theory for C*-algebras, London Mathematical Society Student Texts, vol. 49, Cambridge University Press, Cambridge, 2000. MR 1783408 (2001g:46001)Google Scholar
53.Rosenberg, Jonathan, Algebraic K-theory and its applications, Graduate Texts in Mathematics, vol. 147, Springer-Verlag, New York, 1994. MR 1282290 (95e:19001)CrossRefGoogle Scholar
54.Rudin, Walter, Functional analysis, second ed., International Series in Pure and Applied Mathematics, McGraw-Hill Inc., New York, 1991. MR 1157815 (92k:46001)Google Scholar
55.Sakai, T., Riemannian geometry, American Mathematical Society, Providence, RI, 1996. MR 97f:53001CrossRefGoogle Scholar
56.Schweitzer, Larry B., A short proof that Mn(A) is local if A is local and Fréchet, Internat. J. Math. 3 (1992), no. 4, 581–589. MR 1168361 (93i:46082)CrossRefGoogle Scholar
57.Serre, J.-P., Algèbres de Lie semi-simples complexes, W. A. Benjamin, inc., New York-Amsterdam, 1966. MR 35 #6721Google Scholar
58.Skandalis, Georges, Approche de la conjecture de Novikov par la cohomologie cyclique (d'après A. Connes, M. Gromov et H. Moscovici), Séminaire Bourbaki, Vol. 1990/1991, Astérisque 201–203 (1991), Exp. No. 739, 299–320 (1992), MR 1157846 (93i:57035)Google Scholar
59.Slebarski, Stephen, The Dirac operator on homogeneous spaces and representations of reductive Lie groups. I, Amer. J. Math. 109 (1987), no. 2, 283–301. MR 882424 (89a:22028)CrossRefGoogle Scholar
60.Steinacker, Harold, Quantized gauge theory on the fuzzy sphere as random matrix model, Nuclear Phys. B 679 (2004), no. 1–2, 66–98, arXiv:hep-th/0307075. MR 2033774 (2004k:81409)CrossRefGoogle Scholar
61.Taylor, Michael E., Noncommutative harmonic analysis, Mathematical Surveys and Monographs, vol. 22, American Mathematical Society, Providence, RI, 1986. MR 852988 (88a:22021)CrossRefGoogle Scholar
62.Valtancoli, P., Projectors for the fuzzy sphere, Modern Phys. Lett. A 16 (2001), no. 10, 639–645, arXiv:hep-th/0101189.MR 1833119 (2002m:58012)CrossRefGoogle Scholar
63.Valtancoli, P., Projectors, matrix models and noncommutative monopoles, Internat. J. Modern Phys. A 19 (2004), no. 27, 4641–4657, arXiv:hep-th/0404045. MR 2100603 (2005k:81348)CrossRefGoogle Scholar
64.Warner, Frank W., Foundations of differentiable manifolds and Lie groups, Graduate Texts in Mathematics, vol. 94, Springer-Verlag, New York, 1983, Corrected reprint of the 1971 edition. MR 722297 (84k:58001)CrossRefGoogle Scholar
65.Weaver, N., Lipschitz Algebras, World Scientific, Singapore, 1999. MR 1832645 (2002g:46002)CrossRefGoogle Scholar
66.Wegge-Olsen, N. E., K-theory and C*-algebras, The Clarendon Press Oxford University Press, New York, 1993. MR 1222415 (95c:46116)CrossRefGoogle Scholar