Let X be a locally compact
Hausdorff space, C0(X) the space of continuous real-valued functions on X which
vanish at infinity, and let C0(X) be equipped with the supremum norm. Let
E : C0(X) → C0(X) be a nonnegative projection (x ≧ 0 ⇒ Ex ≧ 0; E2= E) of norm
1. The first theorem states that E(xEy) = E(ExEy) for all x,y ∈ C0(X). Let
X0=⋂{x−1[{0}];x ≧ 0,Ex = 0}. The second theorem states (in part) that
M = E[C0(X)] under the norm and order it inherits from C0(X) is a Banach lattice,
that the mapping x → x∣X0 (= restriction of x to X0) is an isometric vector lattice
homomorphism (= linear map which preserves the lattice operations) of M onto a
subalgebra of C0(X0), and that for t ∈ X0, E(xEy)(t) = (ExEy)(t) for all
x,y ∈ C0(X).
The paper concludes with a characterization of the conditional expectation
operators L1 of a probability space.
