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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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