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Let $B$ and $F$ be compact Hausdorff spaces.

Let $E\to B$ be a fiber bundle with fibre $F$ and structure group $\mathrm{Homeo}(F)$, the group of homeomorphisms of $F$.

I think this induces a fiber bundle $E'$ over $B$ with fiber $C(F,\mathbb C)$, the C*-algebra of continuous functions on $F$, with structure group $\mathrm{Aut}(C(F,\mathbb C))\cong\mathrm{Homeo}(F)$, the group of *-automorphisms of $C(F,\mathbb C)$.

(To be more explicit at this point: my idea is: take a covering of $B$ which trivialises $E$. The transition functions give me a cocycle with values in the structure group $\mathrm{Homeo}(F)$. But, since $\mathrm{Homeo}(F)\cong\mathrm{Aut}(C(F,\mathbb C))$, I get a cocycle with values in $\mathrm{Aut}(C(F,\mathbb C))$, which I'd like to use to glue my new bundle $E'$.)

Let $\Gamma(B,E')$ denote the continuous sections of $E'$. I think pointwise operations turn this into a C*-algebra. Since the fiber $C(F,\mathbb C)$ is commutative, $\Gamma(B,E')$ is commutative as well.

Question: What is the spectrum of $\Gamma(B,E')$?

Example: If $E\cong B\times F$ is the trivial bundle, then $E'\cong B\times C(F,\mathbb C)$ and thus $$\Gamma(B,E')\cong C(B,C(F,\mathbb C))\cong C(B\times F,\mathbb C).$$ This suggests that the spectrum of $\Gamma(B,E')$ is actually $E$.

(This question remained unanswered up to now at math.se.)

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Seems pretty clear, every element $s$ of $\Gamma(B,E')$ gives rise to a continuous function $f_s$ on $E$ given by $$ f_s(e)=s(\pi(e))(e), $$ where $\pi$ is the projection $E\to B$. The other way round, each $f\in C(E)$ gives an element $s_f$ of $\Gamma (B,E')$ by $$ s_f(b)=f|_{\pi^{-1}(b)}. $$ –  doug Feb 19 '11 at 7:28

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