Vol. 28, No. 1, 1969

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 297: 1
Vol. 296: 1  2
Vol. 295: 1  2
Vol. 294: 1  2
Vol. 293: 1  2
Vol. 292: 1  2
Vol. 291: 1  2
Vol. 290: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Subscriptions
Editorial Board
Officers
Special Issues
Submission Guidelines
Submission Form
Contacts
Author Index
To Appear
 
ISSN: 0030-8730
Invariant extensions of linear functionals, with applications to measures and stochastic processes

E. J. McShane, Robert Breckenridge Warfield, Jr. and V. M. Warfield

Vol. 28 (1969), No. 1, 121–142
Abstract

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 Gn1 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 Gk1 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.

Mathematical Subject Classification
Primary: 46.06
Secondary: 28.00
Milestones
Received: 7 November 1967
Published: 1 January 1969
Authors
E. J. McShane
Robert Breckenridge Warfield, Jr.
V. M. Warfield