A global notion of Glimm halving for \cst-algebras is considered which implies that every non-zero quotient of an algebra with this property is antiliminal. We prove subtriviality and selection results for Banach spaces of sections vanishing at infinity of a continuous field of Banach spaces. We use them to prove the global Glimm halving property for strictly antiliminal \cst-algebras with Hausdorff primitive ideal space of finite dimension. This implies that a \cst-algebra $A$ with Hausdorff primitive ideal space of finite dimension must be purely infinite if its simple quotients are purely infinite.