On Quantales and Spectra of C*-Algebras

We study functorial properties of the quantale spectrum MaxA of an arbitrary unital C*-algebra A. In particular we show that the spatialization of MaxA 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 inductive limits 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. We prove that Max is not an equivalence of categories, therefore not providing a generalization of Gelfand duality to the noncommutative case, but we also leave open the possibility of finding such a generalization by proving that Max is a faithful complete invariant of unital C*-algebras.