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Continuous trace \(C^*\)-algebras, gauge groups and rationalization. (English) Zbl 1202.46066
The goal of this paper is to determine the rational homotopy type of the unitaries \(UA_\zeta\) of the section space \(A_\zeta\) of the bundle \(T \times_{PU_n} M_n(\mathbb C) \to X\), where \(\zeta: T \to X\) is a principal \(PU_n\)-bundle over a compact metric space \(X\). The algebra \(A_\zeta\) is a unital continuous trace \(C^*\)-algebra and, in some sense, is the most general such. So the goal of the paper mixes together analysis with algebraic topology in a very attractive way. Now, gauge groups can be expressed as section spaces, so the authors focus on finding the rational types of gauge groups over compact metric spaces. [For finite complexes, cf. Y. Félix and J. Oprea, Proc. Am. Math. Soc. 137, No. 4, 1519–1527 (2009; Zbl 1168.55010)]. For analysts, the paper provides a great deal of the algebraic topological background necessary for the main result. In particular, the authors take great pains to explain how to proceed from results about finite complexes to results about compact metric spaces via inverse limits.

MSC:
46L05 General theory of \(C^*\)-algebras
46J05 General theory of commutative topological algebras
46L85 Noncommutative topology
55P62 Rational homotopy theory
54C35 Function spaces in general topology
55P15 Classification of homotopy type
55P45 \(H\)-spaces and duals
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References:
[1] DOI: 10.1098/rsta.1983.0017 · Zbl 0509.14014
[2] DOI: 10.1007/978-3-540-38117-4
[3] DOI: 10.1112/S0024611500012545 · Zbl 1024.55005
[4] DOI: 10.1007/BF02564376 · Zbl 0167.51704
[5] DOI: 10.2307/2371848 · Zbl 0037.10101
[6] Eilenberg S., Foundations of Algebraic Topology (1952) · Zbl 0047.41402
[7] DOI: 10.1090/S0002-9939-08-09721-9 · Zbl 1168.55010
[8] DOI: 10.2307/1970593 · Zbl 0173.25901
[9] Hilton P., North-Holland Mathematics Studies, in: Localization of Nilpotent Groups and Spaces (1975)
[10] DOI: 10.1090/S0002-9939-1984-0733413-X
[11] DOI: 10.1007/978-1-4684-6254-8
[12] DOI: 10.1090/S0002-9947-08-04477-2 · Zbl 1163.55006
[13] Mac Lane S., Graduate Texts in Mathematics 5, in: Categories for the Working Mathematician (1998) · Zbl 0906.18001
[14] Milnor J. W., Trans. Amer. Math. Soc. 90 pp 272–
[15] DOI: 10.2307/1970615 · Zbl 0163.28202
[16] DOI: 10.1090/surv/060
[17] DOI: 10.1007/BF01168347 · Zbl 0569.55006
[18] DOI: 10.2307/2041968 · Zbl 0346.55020
[19] DOI: 10.2307/1969362 · Zbl 0032.12402
[20] Steenrod N., The Topology of Fiber Bundles (1951) · Zbl 0054.07103
[21] Sullivan D. P., The 1970 MIT notes, K-Monographs in Mathematics 8, in: Geometric Topology: Localization, Periodicity and Galois Symmetry (2005) · Zbl 1078.55001
[22] DOI: 10.1007/978-1-4612-6318-0
[23] DOI: 10.1016/0040-9383(66)90035-8 · Zbl 0163.36702
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.