The Global Glimm Property

Abstract

It is known that a C * -algebra with the Global Glimm Property is nowhere scattered (it has no elementary ideal-quotients), and the Global Glimm Problem asks if the converse holds. We provide a new approach to this long-standing problem by showing that a C * -algebra has the Global Glimm Property if and only if it is nowhere scattered and its Cuntz semigroup is idealfiltered (the Cuntz classes generating a given ideal are downward directed) and has property (V) (a weak form of being sup-semilattice ordered).We show that ideal-filteredness and property (V) are automatic for C *algebras that have stable rank one or real rank zero, thereby recovering the solutions to the Global Glimm Problem in these cases. We also use our approach to solve the Global Glimm Problem for new classes of C * -algebras.