  We study properties of the quantale spectrum Max A of an arbitrary unital
C*-algebra A. In particular we show that the spatialization of Max A with
respect to one of the notions of spatiality in the literature yields the locale
of closed ideals of A when A is commutative. We study under general conditions
functors with this property, in addition requiring that colimits be preserved,
and we conclude in this case that the spectrum of A necessarily coincides with
the locale of closed ideals of the commutative reflection of A. Finally, we
address functorial properties of Max, namely studying (non-)preservation of
limits and colimits. Although Max is not an equivalence of categories,
therefore not providing a direct generalization of Gelfand duality to the
noncommutative case, it is a faithful complete invariant of unital C*-algebras.
