No Access

CONTINUOUS TRACE C*-ALGEBRAS, GAUGE GROUPS AND RATIONALIZATION

JOHN R. KLEIN

Department of Mathematics, Wayne State University, Detroit MI 48202, USA

Search for more papers by this author

CLAUDE L. SCHOCHET

Department of Mathematics, Wayne State University, Detroit MI 48202, USA

Search for more papers by this author

, and

SAMUEL B. SMITH

Department of Mathematics, Saint Joseph's University, Philadelphia PA 19131, USA

Search for more papers by this author

https://doi.org/10.1142/S179352530900014XCited by:8 (Source: Crossref)

Abstract

Let ζ be an n-dimensional complex matrix bundle over a compact metric space X and let A_ζ denote the C*-algebra of sections of this bundle. We determine the rational homotopy type as an H-space of UA_ζ, the group of unitaries of A_ζ. The answer turns out to be independent of the bundle ζ and depends only upon n and the rational cohomology of X. We prove analogous results for the gauge group and the projective gauge group of a principal bundle over a compact metric space X.

Keywords:

AMSC: 46J05, 46L85, 55P62, 54C35, 55P15, 55P45

References

M. F. Atiyah and R. Bott, Philos. Trans. Roy. Soc. London Ser. A 308, 523 (1983). Crossref, Google Scholar
A. K. Bousfield and D. M. Kan , Homotopy Limits, Completions, and Localizations , Lecture Notes in Mathematics 304 ( Springer , 1972 ) . Crossref, Google Scholar
M. C. Crabb and W. A. Sutherland, Proc. London. Math. Soc. 81, 747 (2000), DOI: 10.1112/S0024611500012545. Crossref, Google Scholar
C. J. Curjel, Comment. Math. Helv. 43, 1 (1968), DOI: 10.1007/BF02564376. Crossref, Google Scholar
C. H. Dowker, Amer. J. Math. 69, 200 (1947), DOI: 10.2307/2371848. Crossref, Google Scholar
S. Eilenberg and N. Steenrod , Foundations of Algebraic Topology ( Princeton Univ. Press , 1952 ) . Crossref, Google Scholar
Y. Félix and J. Oprea, Proc. Amer. Math. Soc. 137, 1519 (2009). Crossref, Google Scholar
D. H. Gottlieb, Ann. Math. 87, 42 (1968), DOI: 10.2307/1970593. Crossref, Google Scholar
P. Hilton , G. Mislin and J. Roitberg , Localization of Nilpotent Groups and Spaces , North-Holland Mathematics Studies ( North-Holland , 1975 ) . Google Scholar
P. J. Kahn, Proc. Amer. Math. Soc. 90, 599 (1959), DOI: 10.2307/2045037. Crossref, Google Scholar
A. Lundell and S. Weingram , The Topology of CW Complexes , University Series in Higher Mathematics ( Van Nostrand Reinhold Co. , 1969 ) . Crossref, Google Scholar
G. Luptonet al., Trans. Amer. Math. Soc. 361, 267 (2009), DOI: 10.1090/S0002-9947-08-04477-2. Crossref, Google Scholar
S. Mac Lane , Categories for the Working Mathematician , Graduate Texts in Mathematics 5 ( Springer , 1998 ) . Google Scholar
J. W. Milnor, Trans. Amer. Math. Soc. 90, 272 (1959), DOI: 10.2307/1993204. Google Scholar
J. W. Milnor and J. C. Moore, Ann. Math. 81, 211 (1965), DOI: 10.2307/1970615. Crossref, Google Scholar
I. Raeburn and D. P. Williams , Morita Equivalence and Continuous-Trace C* -Algebras , Math. Surveys and Monographs 60 ( Amer. Math. Soc. , 1998 ) . Crossref, Google Scholar
H. Scheerer, Manuscripta Math. 51, 63 (1985), DOI: 10.1007/BF01168347. Crossref, Google Scholar
R. Schön, Proc. Amer. Math. Soc. 62, 165 (1977). Crossref, Google Scholar
E. Spanier, Ann. Math. 50, 203 (1949), DOI: 10.2307/1969362. Crossref, Google Scholar
N. Steenrod , The Topology of Fiber Bundles ( Princeton Univ. Press , 1951 ) . Crossref, Google Scholar
D. P. Sullivan , Geometric Topology: Localization, Periodicity and Galois Symmetry , The 1970 MIT notes, K-Monographs in Mathematics 8 ( Springer , 2005 ) . Crossref, Google Scholar
R. Thom, L'homologie des espaces fonctionnels, Colloque de topologie algébrique, Louvain, 1956, Georges Thone, Liège, 1957, pp. 29–39 . Google Scholar
G. Whitehead , Elements of Homotopy Theory , Graduate Texts in Mathematics 61 ( Springer , 1978 ) . Crossref, Google Scholar
R. Wood, Topology 4, 371 (1966), DOI: 10.1016/0040-9383(66)90035-8. Crossref, Google Scholar