Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-24T23:11:21.308Z Has data issue: false hasContentIssue false

Tight Riesz groups

Published online by Cambridge University Press:  09 April 2009

R. J. Loy
Affiliation:
Carleton University Ottawa
J. B. Miller
Affiliation:
Trent UniversityPeterborough
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The theory of partially ordered topological groups has received little attention in the literature, despite the accessibility and importance in analysis of the group Rm. One obstacle in the way of a general theory seems to be, that a convenient association between the ordering and the topology suggests that the cone of all strictly positive elements be open, i.e. that the topology be at least as strong as the open-interval topology U; but if the ordering is a lattice ordering and not a full ordering then U itself is already discrete. So to obtain in this context something more interesting topologically than the discrete topology and orderwise than the full order, one must forego orderings which make lattice-ordered groups: in fact, the partially ordered group must be an antilattice, that is, must admit no nontrivial meets or joins (see § 2, 10°).

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1972

References

Birkhoff, G. (1967): Lattice theory (Amer. Math. Soc., Providence, 3rd ed.).Google Scholar
Dugundji, J. (1966): Topology (Allyn and Bacon, Boston).Google Scholar
Fuchs, L. (1963): Partially ordered algebraic systems (Pergamon, Oxford).Google Scholar
Fuchs, L. (1965): Riesz groups, Annali della Scuola Normale Superiore di Pica, Serie III, 19, 134.Google Scholar
Fuchs, L. (1966): Riesz vector spaces and Riesz algebras (Queen's papers in pure and applied mathematics, Queen's University, Kingston).Google Scholar
Hewitt, E. and Ross, K. A. (1963): Abstract harmonic analysis, I (Springer, Berlin).Google Scholar
Mackey, G. W. (1948): The Laplace transform for locally compact abelian groups, Proc. Nat. Acad. Sci. U.S.A. 34, 156162.CrossRefGoogle ScholarPubMed
Miller, J. B. (1970): Higher derivations on Banach algebras, Amer. J. of Math. 92, 301331.CrossRefGoogle Scholar
Miller, J. B. (1972): Tight Riesz groups and the Stone-Weierstrass theorem, J. Aust. Math. Soc. (to appear).Google Scholar
Peressini, A. L. (1967): Ordered topological vector spaces (Harper and Row, New York).Google Scholar
Riesz, F. (1940): Sur quelques notions fondamentales dans la théorie générale des opérations linéaires, Annals Math. 41, 174206.CrossRefGoogle Scholar