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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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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Additional Information

Gregory Lupton
Affiliation: Department of Mathematics, Cleveland State University, Cleveland, Ohio 44115
Email: G.Lupton@csuohio.edu

N. Christopher Phillips
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222
Email: ncp@darkwing.uoregon.edu

Claude L. Schochet
Affiliation: Department of Mathematics, Wayne State University, Detroit, Michigan 48202
Email: claude@math.wayne.edu

Samuel B. Smith
Affiliation: Department of Mathematics, Saint Joseph’s University, Philadelphia, Pennsylvania 19131
Email: smith@sju.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04477-2
PII: S 0002-9947(08)04477-2
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.