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Transactions of the American Mathematical Society

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Banach algebras and rational homotopy theory

Authors: Gregory Lupton, N. Christopher Phillips, Claude L. Schochet and Samuel B. Smith
Journal: Trans. Amer. Math. Soc. 361 (2009), 267-295
MSC (2000): Primary 46J05, 46L85, 55P62, 54C35, 55P15, 55P45
Published electronically: August 14, 2008
MathSciNet review: 2439407
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Abstract: Let $ A$ be a unital commutative Banach algebra with maximal ideal space $ \operatorname{Max}(A).$ We determine the rational H-type of $ \operatorname{GL}_n (A),$ the group of invertible $ n \times n$ matrices with coefficients in $ A,$ in terms of the rational cohomology of $ \operatorname{Max} (A).$ We also address an old problem of J. L. Taylor. Let $ \operatorname{Lc}_n (A)$ denote the space of ``last columns'' of $ \operatorname{GL}_n (A).$ We construct a natural isomorphism

$\displaystyle {\Check{H}}^s (\operatorname{Max} (A); \mathbb{Q} ) \cong \pi_{2 n - 1 - s} (\operatorname{Lc}_n (A)) \otimes \mathbb{Q} $

for $ n > \frac{1}{2} s + 1$ which shows that the rational cohomology groups of $ \operatorname{Max} (A)$ are determined by a topological invariant associated to $ A.$ As part of our analysis, we determine the rational H-type of certain gauge groups $ F (X, G)$ for $ G$ a Lie group or, more generally, a rational H-space.

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  • 1. J. F. Adams, On the non-existence of elements of Hopf invariant one, Ann. Math. 72 (1960), 20-104. MR 25:4530
  • 2. M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), 523-615. MR 85k:14006
  • 3. I. Berstein and T. Ganea, Homotopical nilpotency, Illinois J. Math. 5 (1961), 99-130. MR 23:A3573
  • 4. F. F. Bonsall and J. Duncan, Complete normed algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 80, Springer-Verlag, New York, Heidelberg, 1973. MR 54:11013
  • 5. A. Borel, Topology of Lie groups and characteristic classes, Bull. Amer. Math. Soc. 61 (1955), 397-432. MR 17:282b
  • 6. L. G. Brown, P. Green, and M. A. Rieffel, Stable isomorphism and strong Morita equivalence of $ C^*$-algebras, Pacific J. Math. 71 (1977), 349-363. MR 57:3866
  • 7. G. Corach and A. R. Larotonda, Stable range in Banach algebras, J. Pure Appl. Algebra 32 (1984), 289-300. MR 86g:46070
  • 8. -, A stabilization theorem for Banach algebras, J. Algebra 101 (1986), 433-449. MR 87h:46103
  • 9. C. J. Curjel, On the $ H$-space structures of finite complexes, Comment. Math. Helv. 43 (1968), 1-17. MR 37:6929
  • 10. A. M. Davie, Homotopy in Fréchet algebras, Proc. London Math. Soc. (3) 23 (1971), 31-52. MR 45:5756
  • 11. J. Dixmier and A. Douady, Champs continus d'espaces hilbertiens et de $ C^*$-algèbres, Bull. Soc. Math. France 91 (1963), 227-284. MR 29:485
  • 12. S. Eilenberg and N. E. Steenrod, Foundations of algebraic topology, Princeton University Press, Princeton, 1952. MR 14:398b
  • 13. Y. Félix, S. Halperin, and J.-C. Thomas, Rational homotopy theory, Graduate Texts in Mathematics, vol. 205, Springer-Verlag, New York, 2001. MR 2002d:55014
  • 14. D. Handelman, $ K_{0}$ of von Neumann and AF C* algebras, Quart. J. Math. Oxford Ser. (2) 29 (1978), 427-441. MR 81c:46049
  • 15. P. Hilton, G. Mislin, and J. Roitberg, Localization of nilpotent groups and spaces, North-Holland Publishing Co., Amsterdam, 1975, North-Holland Mathematics Studies, No. 15, Notas de Matemática, No. 55 [Notes on Mathematics, No. 55]. MR 57:17635
  • 16. I. M. James, On $ H$-spaces and their homotopy groups, Quart. J. Math. Oxford Ser. (2) 11 (1960), 161-179. MR 24:A2966
  • 17. J. L. Kelley, General topology, Van Nostrand Reinhold, New York, Cincinnati, Toronto, London, Melbourne, 1955. MR 16:1136c
  • 18. A. T. Lundell and S. Weingram, The topology of CW complexes, Van Nostrand Reinhold Company, New York, 1969.
  • 19. R. J. Milgram, The bar construction and abelian $ H$-spaces, Illinois J. Math 11 (1967), 242-250. MR 34:8404
  • 20. J. Milnor, The geometric realization of a semi-simplicial complex, Ann. Math. 65 (1957), 357-362. MR 18:815d
  • 21. -, On spaces having the homotopy type of CW-complex, Trans. Amer. Math. Soc. 90 (1959), 272-280. MR 20:6700
  • 22. G. J. Murphy, $ C^*$-algebras and operator theory, Academic Press, Boston, San Diego, New York, London, Sydney, Tokyo, Toronto, 1990. MR 91m:46084
  • 23. N. C. Phillips, Equivariant K-theory and freeness of group actions on $ C^*$-algebras, Springer-Verlag Lecture Notes in Math. no. 1274, Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1987. MR 89k:46086
  • 24. M. A. Rieffel, Dimension and stable rank in the $ K$-theory of $ C^*$-algebras, Proc. London Math. Soc. (3) 46 (1983), 301-333. MR 84g:46085
  • 25. -, The homotopy groups of the unitary groups of noncommutative tori, J. Operator Theory 17 (1987), 237-254. MR 88f:22018
  • 26. H. Scheerer, On rationalized H- and co-H-spaces. With an appendix on decomposable H- and co-H-spaces, Manuscripta Math. 51 (1985), 63-87. MR 88k:55007
  • 27. E. H. Spanier, Algebraic topology, McGraw-Hill, New York, San Francisco, St. Louis, Toronto, London, Sydney, 1966. MR 35:1007
  • 28. J. Stasheff, $ H$-spaces from a homotopy point of view, Lecture Notes in Mathematics, vol. 161, Springer-Verlag, Berlin, New York, 1970. MR 42:5261
  • 29. J. L. Taylor, Banach algebras and topology, Algebras in analysis (Proc. Instructional Conf. and NATO Advanced Study Inst., Birmingham, 1973), Academic Press, London, 1975, pp. 118-186. MR 54:5837
  • 30. A. B. Thom, Connective $ E$-theory and bivariant homology for $ C^*$-algebras, Ph.D. thesis, U. Münster, 2003.
  • 31. R. Thom, L'homologie des espaces fonctionnels, Colloque de topologie algébrique, Louvain, 1956, Georges Thone, Liège, 1957, pp. 29-39. MR 19:669h
  • 32. K. Thomsen, Nonstable $ K$-theory for operator algebras, $ K$-Theory 4 (1991), 245-267. MR 92h:46102
  • 33. G. W. Whitehead, Elements of homotopy theory, Graduate Texts in Mathematics, vol. 61, Springer-Verlag, New York, 1978. MR 80b:55001

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Additional Information

Gregory Lupton
Affiliation: Department of Mathematics, Cleveland State University, Cleveland, Ohio 44115

N. Christopher Phillips
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222

Claude L. Schochet
Affiliation: Department of Mathematics, Wayne State University, Detroit, Michigan 48202

Samuel B. Smith
Affiliation: Department of Mathematics, Saint Joseph’s University, Philadelphia, Pennsylvania 19131

Keywords: Commutative Banach algebra, maximal ideal space, general linear group, space of last columns, rational homotopy theory, function space, rational H-space, gauge groups
Received by editor(s): April 17, 2006
Received by editor(s) in revised form: December 19, 2006
Published electronically: August 14, 2008
Additional Notes: The research of the second author was partially supported by NSF grant DMS 0302401.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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