  We study Fell bundles on groupoids from the viewpoint of quantale theory.
Given any saturated upper semicontinuous Fell bundle $\pi:E\to G$ on an \'etale
groupoid $G$ with $G_0$ locally compact Hausdorff, equipped with a suitable
completion C*-algebra $A$ of its convolution algebra, we obtain a map of
involutive quantales $p:\mathrm{Max}\ A\to\Omega(G)$, where $\mathrm{Max}\ A$
consists of the closed linear subspaces of $A$ and $\Omega(G)$ is the topology
of $G$. We study various properties of $p$ which mimick, to various degrees,
those of open maps of topological spaces. These are closely related to
properties of $G$, $\pi$, and $A$, such as $G$ being Hausdorff, principal, or
topological principal, or $\pi$ being a line bundle. Under suitable conditions,
which include $G$ being Hausdorff, but without requiring saturation of the Fell
bundle, $A$ is an algebra of sections of the bundle if and only if it is the
reduced C*-algebra $C_r^*(G,E)$. We also prove that $\mathrm{Max}\ A$ is stably
Gelfand. This implies the existence of a pseudogroup $\mathcal{I}_B$ and of an
\'etale groupoid $\mathfrak B$ associated canonically to any sub-C*-algebra
$B\subset A$. We study a correspondence between Fell bundles and
sub-C*-algebras based on these constructions, and compare it to the
construction of Weyl groupoids from Cartan subalgebras.
