The Riesz Space of Minimal Usco Maps

We consider upper semi-continuous compact-valued (usco) maps with values in a Banach lattice. Recently, it was shown that the space M(X, Y) of minimal upper semi-continuous compact-valued maps from a topological space X into a metrizable topological vector space Y is a vector space which contains the space C(X, Y) of continuous functions from X into Y as a linear subspace. In this paper, we consider the situation when the range space is a Banach lattice E. In this case, C(X, E) is a Riesz space with respect to the usual pointwise ordering. We show that M(X, E) is equipped in a natural way with a partial order that extends the order on C(X, E). With respect to this order, M(X, E) is an Archimedean Riesz space. Moreover, if E has compact order intervals, then M(X, E) is Dedekind complete. An application is made to the characterisation of the Dedekind completion of C(X, E).