Suppose that X is a compact
Hausdorff space and that F is a closed (linear) sublattice of C(X). We characterize
those sublattices F that are the ranges of (linear) Iattice projections on C(X): there
is a lattice projection of C(X) onto F if and only if there is a closed subset Y of X
such that F is lattice isomorphic to C(Y ) under the restriction mapping
f → f|Y(f ∈ F). Examples are given to show that this theorem cannot be
substantially improved without imposing additional conditions either on X or on the
sublattice F. If X is a stonian space, then a closed sublattice F of C(X) is
the range of a lattice projection exactly when it is the range of a positive
projection.
