Hostname: page-component-84b7d79bbc-g78kv Total loading time: 0 Render date: 2024-07-28T05:33:39.712Z Has data issue: false hasContentIssue false
  • English
  • Français

Spaces of sections of Banach algebra bundles

Published online by Cambridge University Press:  04 April 2012

Emmanuel Dror Farjoun  and
Claude L. Schochet
Emmanuel Dror Farjoun
Affiliation:
Department of Mathematics, Hebrew University of Jerusalem, Jerusalem 91904, Israel, farjoun@huji.ac.il
Claude L. Schochet
Affiliation:
Department of Mathematics, Wayne State University, Detroit MI 48202, claude@math.wayne.edu
Get access

Abstract

Suppose that B is a G-Banach algebra over = ℝ or ℂ X is a finite dimensional compact metric space, ζ : P → X is a standard principal G-bundle, and Aζ = Γ(X,P ×GB) is the associated algebra of sections. We produce a spectral sequence which converges to π*(GLoAζ) with

A related spectral sequence converging to K*+1(Aζ) (the real or complex topological K-theory) allows us to conclude that if B is Bott-stable, (i.e., if π*(GLoB) → K*+1(B) is an isomorphism for all * > 0) then so is Aζ.

Keywords

general linear group of a Banach algebraspectral sequenceslocalizationBott-stableK-theory for Banach algebrasunstable K-theory
Type
Research Article
Information
Journal of K-Theory , Volume 10 , Issue 2 , October 2012 , pp. 279 - 298
Copyright
Copyright © ISOPP 2012

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

REFERENCES

1.Adams, J. F., Vector fields on spheres, Annals. of Math. (2) 75 (1962), 603–632.Google Scholar
2.Atiyah, M. F. and Segal, G., Twisted K-theory and cohomology, Inspired by Chern, S. S., 5–43, Nankai Tracts Math. 11, World Sci. Publ., Hackensack, NJ, 2006.Google Scholar
3.Bott, R., The stable homotopy of the classical groups, Ann. of Math. 70 (1959), 313–337.Google Scholar
4.Bousfield, A. K. and Kan, D. M., Homotopy Limits, Completions, and Localizations, Lecture Notes in Mathematics 304, Springer-Verlag, New York, 1972.CrossRefGoogle Scholar
5.Dadarlat, M., Continuous fields of C*-algebras over finite dimensional spaces, Adv. Math. 222 (2009), no. 5, 1850–1881.CrossRefGoogle Scholar
6.Dadarlat, M., The C *-algebra of a vector bundle, arXiv:1004.1722.Google Scholar
7.Eilenberg, S. and Steenrod, N., Foundations of Algebraic Topology, Princeton University Press, Princeton, 1952.Google Scholar
8.Farjoun, E. Dror, Bousfield-Kan completion of homotopy limits, Topology 42 (2003), 1083–1099.Google Scholar
9.Federer, H., A study of function spaces by spectral sequences, Trans. Amer. Math. Soc. 82 (1956), 340–361.Google Scholar
10.Hilton, P., Mislin, G., and Roitberg, J., Localization of nilpotent groups and spaces, North-Holland Publishing Co., Amsterdam, 1975, North-Holland Mathematics Studies 15, Notas de Matemática 55 [Notes on Mathematics 55].Google Scholar
11.Karoubi, M., Algèbres de Clifford et K-théorie, Ann. Sci. École Norm. Sup. (4) 1 1968, 161–270.Google Scholar
12.Klein, J., Schochet, C., and Smith, S., Continuous trace C*-algebras, gauge groups and rational homotopy, J. Topology and Analysis 1 (2009), 261–288.CrossRefGoogle Scholar
13.Legrand, A., Homotopie des espaces de sections, Lecture Notes in Mathematics 941, Springer-Verlag, New York, 1982, vi + 132 pp.Google Scholar
14.Lupton, G., Phillips, N. C., Schochet, C. L., S.B. Smith, Banach algebras and rational homotopy theory, Trans. Amer. Math. Soc. 361 (2009), 267–295.Google Scholar
15.Mac, S. Lane, Homology, Die Grund. der Math. 114, Springer Verlag, New York, 1963.Google Scholar
16.McCleary, J., A User's Guide to Spectral Sequences, 2nd. ed., Cambridge Stud. Adv. Math. 58, Cambridge University Press, Cambridge, 2001. xvi + 561 pp.Google Scholar
17.Milnor, J. W. and Stasheff, J. D., Characteristic Classes, Annals of Math. Studies 76, Princeton University Press, 1974.Google Scholar
18.Moore, J. C., Comparaison de la bar-construction à la construction W et aux complexes K(π,η), Seminaire H. Cartan 7, no. 2 (Paris, 1954–1955), Exposé 13. available at numdam.org.Google Scholar
19.Phillips, N. C., private communication.Google Scholar
20.Rosenberg, J., Homological invariants of extensions of C*-algebras, Operator algebras and applications, Part 1 (Kingston, Ont., 1980), pp. 35–75, Proc. Sympos. Pure Math. 38, Amer. Math. Soc, Providence, RI, 1982.Google Scholar
21.Smith, S. B., A based Federer spectral sequence and the rational homotopy of function spaces, Manuscripta Math. 93 (1997), 59–66.Google Scholar
22.Thom, R., L'homologie des espaces fonctionnels, Colloque de topologie algébrique, Louvain, 1956, Georges Thone, Liège, 1957, pp. 29–39.Google Scholar
23.Thomsen, K., Nonstable K-theory for operator algebras, K-Theory 4 (1991), 245–267.Google Scholar
24.Wood, R., Banach algebras and Bott periodicity, Topology 4 (1966), 371–389.Google Scholar