Vol. 17, No. 1, 1966

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Nonnegative projections on C0(X)

Galen Lathrop Seever

Vol. 17 (1966), No. 1, 159–166

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 = {x1[{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 xX0 (= 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.

Mathematical Subject Classification
Primary: 47.25
Received: 23 October 1964
Published: 1 April 1966
Galen Lathrop Seever