Universal spaces of subdifferentials of sublinear operators ranging in the cone of bounded lower semicontinuous functions

We study Fréchet’s problem of the universal space for the subdifferentials ∂P of continuous sublinear operators P: V → BC(X)∼ which are defined on separable Banach spaces V and range in the cone BC(X)∼ of bounded lower semicontinuous functions on a normal topological space X. We prove that the space of linear compact operators Lc(ℓ2, C(βX)) is universal in the topology of simple convergence. Here ℓ2 is a separable Hilbert space, and βX is the Stone-Ĉech compactification of X. We show that the images of subdifferentials are also subdifferentials of sublinear operators.