Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-25T13:42:39.968Z Has data issue: false hasContentIssue false
  • English
  • Français

Minimal decompositions in partially ordered normed vector spaces

Published online by Cambridge University Press:  24 October 2008

A. J. Ellis
A. J. Ellis
Affiliation:
University College of Swansea
Get access Rights & Permissions [Opens in a new window]

Extract

In this paper we study partially ordered vector spaces X whose positive cone K possesses a base which defines a norm in X. A positive decomposition x = y − z of the element x is said to be minimal if ‖x‖ = ‖y‖ + ‖z‖. We proved in (6) that the property that every element of X has a unique minimal decomposition is equivalent to an intersection property for homothetic translates of the base. Section 2 of the present paper analyses this intersection property in much more detail and discusses possible generalizations of it.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 64 , Issue 4 , October 1968 , pp. 989 - 1000
Copyright
Copyright © Cambridge Philosophical Society 1968

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Bishop, E. and Phelps, R. R. The support functionals of a convex set. Proceedings of Symposia in Pure Mathematics VII (Convexity), pp. 27–35 (Providence, R.I., 1963).Google Scholar
(2)Day, M. M.Normed linear spaces (Springer-Verlag, Berlin, 1958).CrossRefGoogle Scholar
(3)Edwards, D. A.On the homeomorphic affine embedding of a locally compact cone into a Banach dual space endowed with the vague topology. Proc. London Math. Soc. (3), 14 (1964), 399–414.CrossRefGoogle Scholar
(4)Effros, E. G.Structure in simplexes. Acta Math. 117 (1967), 103–121.CrossRefGoogle Scholar
(5)Ellis, A. J.The duality of partially ordered normed linear spaces. J. London Math. Soc. 39 (1964), 730–744.CrossRefGoogle Scholar
(6)Ellis, A. J.An intersection property for state spaces. J. London Math. Soc. 43 (1968), 173–76.CrossRefGoogle Scholar
(7)Grothendieck, A.Un résultat sur le dual d'une C *-algebre. J. Math. Pures Appl. (9), 36 (1957), 97–108.Google Scholar
(8)Kendall, D. G.Simplexes and vector lattices. J. London Math. Soc. 37 (1962), 365–371.CrossRefGoogle Scholar
(9)Klee, V. L. JrSeparation properties of convex cones. Proc. Amer. Math. Soc. 6 (1955), 313–318.CrossRefGoogle Scholar
(10)Klee, V. L. JrExtremal structure of convex sets. I. Arch. Math. (Basel) 8 (1957), 234–240.CrossRefGoogle Scholar
(11)Klee, V. L. JrExtremal structure of convex sets. II. Math. Z. 69 (1958), 90–104.CrossRefGoogle Scholar
(12)Prosser, R. T.On the ideal structure of operator algebras. Mem. Amer. Math. Soc. 45 (1963).Google Scholar
(13)Sakai, S.A characterization of W *-algebras. Pacific J. Math. 6 (1956), 763–773.CrossRefGoogle Scholar