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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The positive fixed points of Banach lattices

Author: Bruce Christianson
Journal: Proc. Amer. Math. Soc. 107 (1989), 255-260
MSC: Primary 46B30
MathSciNet review: 990419
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Abstract: Let $ Z$ be a Banach lattice endowed with positive cone $ C$ and an order-continuous norm $ \vert\vert \cdot \vert\vert$. Let $ G$ be a left-amenable semigroup of positive linear endomorphisms of $ Z$. Then the positive fixed points $ {C_0}$ of $ Z$ under $ G$ form a lattice cone, and their linear span $ {Z_0}$ is a Banach lattice under an order-continuous norm $ \vert\vert \cdot \vert{\vert _0}$ which agrees with $ \vert\vert \cdot \vert\vert$ on $ {C_0}$. A counterexample shows that under the given conditions $ {Z_0}$ need not contain all the fixed points of $ Z$ under $ G$, and need not be a sublattice of $ (Z,C)$. The paper concludes with a discussion of some related results.

References [Enhancements On Off] (What's this?)

  • [1] M. M. Day, Normed linear spaces, Third Edition, Ergeb. Math. Grenzgeb. 21 (1973), Springer-Verlag, Berlin. MR 0344849 (49:9588)
  • [2] C. M. Edwards and M. A. Gerzon, Monotone convergence in partially ordered vector spaces, Ann. Inst. H. Poincaré Phys. Théor. (A)12 (1970), 323-328. MR 0268644 (42:3541)
  • [3] H. H. Schaefer, Banach lattices and positive operators, Grundlehren Math. Wiss. 215 (1974), Springer-Verlag, Berlin. MR 0423039 (54:11023)
  • [4] G. Winkler, Choquet order and simplices with applications in probabilistic models, Lecture Notes in Math. no. 1145 (1985), Springer-Verlag, Berlin. MR 808401 (88f:46019)

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Article copyright: © Copyright 1989 American Mathematical Society

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