A theorem is proved
slightly stronger than the following. Let G be a set of order-preserving linear
operators on a partially-ordered real linear space X, for which there exist sets
G = Gn⊇ Gn−1⊇⋯⊇ G0 with G0 commutative and such that for k = 1,⋯,n, x in
X, g1 and g2 in Gk there exist h1 and h2 in Gk−1 satisfying h1g1g2(x) = h2g2g1(x).
If S is a G-invariant subspace such that for all x in X there is an s in S satisfying
s ≧ x, and f0 is a G-invariant positive linear functional on S, then f0 extends to a
G-invariant positive linear functional on X. This is used to construct a generalized
form of the Banach limit, an ergodic measure on compact Hausdorff spaces, a
stationary extension of a relatively stationary stochastic process xt(0 ≦ t ≦ a)
with values in an arbitrary space, and a generalization to arbitrary linear
spaces of Krein’s extension theorem for positive-definite complex-valued
functions.
