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Proceedings of the American Mathematical Society
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.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1989-0990419-8
PII: S 0002-9939(1989)0990419-8
Article copyright: © Copyright 1989 American Mathematical Society