Exposed faces of dual cones and peak-set criteria for function spaces

This paper gives conditions for a closed subcone of a weak¹ closed cone in a Banach dual space to be exposed by a weak¹ continuous linear functional. This set-up is applied to the study of complex-valued and vector-valued continuous function spaces on a compact Hausdorff space to deduce peak-set criteria. In the case of complex-valued function spaces peak-set conditions are given in terms of gages on certain cones of complex measures. These conditions are shown to imply various known peak-set criteria involving annihilating measures.