Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-30T05:38:04.820Z Has data issue: false hasContentIssue false
  • English
  • Français

Existence of Invariant Weak Units in Banach Lattices: Countably Generated Left Amenable Semigroup of Operators

Published online by Cambridge University Press:  20 November 2018

K. Prabaharan
K. Prabaharan*
Affiliation:
Department of Mathematics Ohio State University Columbus, Ohio 43210, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

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.

Let Σ be a countably generated left amenable semigroup and ﹛Tσ|σ ∈ Σ﹜ be a representation of Σ as a semigroup of positive linear operators on a weakly sequentially complete Banach lattice E with a weak unit e. It is assumed Tσ are uniformly bounded. It is shown that a necessary and sufficient condition for the existence of a weak unit invariant under ﹛Tσ | σ ∈ Σ﹜ is that inf σ∈Σ H(Tσe) > 0 for all nonzero H in the positive dual cone of E.

Keywords

46B3047B5528D15
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 45 , Issue 6 , 01 December 1993 , pp. 1299 - 1312
Copyright
Copyright © Canadian Mathematical Society 1993

References

1. Akcoglu, M.A. and Sucheston, L., An ergodic theorem on Banach lattices, Israel J. Math. 51(1985), 208–222.Google Scholar
2. Blum, J.R. and Friedman, N., On invariant measures for classes of transformations, Z. Wahrscheinlichkeitstheorie verw. Geb. 8(1967), 301–305.Google Scholar
3. Brunei, A. and Sucheston, L., Sur I ‘existence d'éléments invariants dans les treillis de Banach, C.R. Acad. Sci. Paris (1)300(1985), 59–62.Google Scholar
4. Calderon, A., Sur les measures invariantes, C.R. Acad. Sci. Paris 240(1955), 1960.1962.Google Scholar
5. Day, M., Normed linear spaces. 2nd printing corrected, Academic Press, New York, 1962.Google Scholar
6. Day, M., Amenable semigroups, Illinois J. Math. 1(1957), 509–544.Google Scholar
7. Dixmier, J., Les moynes invariantes dans les semigroupes et leurs aplications, Acta Sci. Math. Szeged. 12(1950), 213–227.Google Scholar
8. Dowker, Y.N., On measurable transformations infinite measure spaces, Ann. of Math. (2) 62(1955), 504–516.Google Scholar
9. Dowker, Y.N., Sur les applications measurable, C.R. Acad. Sci. Paris 242(1956), 329–331.Google Scholar
10. Granirer, E., On finite equivalent invariant measures for semigroups of transformations, Duke Math. J. 38(1971), 395–408.Google Scholar
11. Greenleaf, F., Invariant means on topological groups, Math. Studies, Princeton, 1969.Google Scholar
12. Hajian, A. and Kakutani, S., Weakly wandering sets and invariant measures, Trans. Amer. Math. Soc. 110(1964), 135–151.Google Scholar
13. Krengel, U., Ergodic Theorems, de Gruyter studies in mathematics, Berlin, 1985.Google Scholar
14. Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces II-Function Spaces, Springer-Verlag, Berlin- Heidelberg-New York, 1979.Google Scholar
15. Nakano, H., Modulared semi-ordered linear spaces, Maruzen, Tokyo, 1950.Google Scholar
16. Sachdeva, U., On finite invariant measure for semigroups of operators, Canad. Math. Bull. (2) 14(1971), 197–206.Google Scholar
17. Shields, P.C., Invariant elements for positive contractions on a Banach lattice, Math. Z. 96(1967), 189–195.Google Scholar
18. Sucheston, L., On existence of finite invariant measures, Math. Z. 86(1964), 327–336.Google Scholar