Continuous Trace C*-Algebras, Gauge Groups and Rationalization

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.

Spaces of sections of Banach algebra bundles

“…This is a consequence, noted in [8], of the basic Fibre Lemma of Bousfield and Kan [4]. Next, assume that X is a finite-dimensional compact metric space, write it as an inverse limit of finite CW-complexes, and use our first step results together with limit techniques of [12] to complete the argument.…”

“…X j denote the resulting structure maps. Then, as shown in [12], for j sufficiently large, there are maps 5 f j W X j ! BG and a homotopy-commuting diagram of the form…”

“…The Corollary was established for X compact metric, B D M n .C/, P D Q and trivial, so that A D F .X;M n .C//, by Lupton, Phillips, Schochet, and Smith [14] and in general by Klein, Schochet, and Smith [12]. The idea of using a spectral sequence to compute .F .X;Y // goes back to Federer [8], and note particularly the work of S. Smith [21].…”

“…Our convention is to discard all terms of non-positive degree. Proof: Part (1) of the Theorem is established in ( [12], page 264, Addendum E). Then the results of [14] on the rational homotopy of function spaces of this type imply Part (2).…”

“…Corollary 1.1 ( [14], [12]) With the above notation, if A is a unital continuoustrace C -algebra over C and P D Q then E 2 D E 1 and…”