Transactions of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Banach algebras and rational homotopy theory

Author(s): Gregory Lupton; N. Christopher Phillips; Claude L. Schochet; Samuel B. Smith
Journal: Trans. Amer. Math. Soc. 361 (2009), 267-295.
MSC (2000): Primary 46J05, 46L85, 55P62, 54C35, 55P15, 55P45
Posted: August 14, 2008
MathSciNet review: 2439407
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: Let 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 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

As part of our analysis, we determine the rational H-type of certain gauge groups

for

a Lie group or, more generally, a rational H-space.

References:

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 -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 -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 -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 -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 -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, -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 -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 -theory of -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, -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 -theory and bivariant homology for -algebras, Ph.D. thesis, U. Münster, 2003. math.uni-muenster.de/inst/sfb/about/publ/thom.html
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 -theory for operator algebras, -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

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|