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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the semiring $ L\sp +(C\sb 0(X))$


Author: Jor-Ting Chan
Journal: Proc. Amer. Math. Soc. 112 (1991), 171-174
MSC: Primary 46E25; Secondary 06F25, 46E15
MathSciNet review: 1037205
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Abstract: Let $ X$ and $ Y$ be locally compact Hausdorff spaces. Let $ {C_0}(X)$ (resp. $ {C_0}(Y)$) denote the Banach space of all continuous functions on $ X$ vanishing at infinity on $ X$ (resp. $ Y$) and $ {L^ + }({C_0}(X))$ (resp. $ {L^ + }({C_0}(Y))$) the semiring of positive operators on $ {C_0}(X)$ (resp. $ {C_0}(Y)$). We prove that if there exists a semiring isomorphism $ \varphi $ from $ {L^ + }({C_0}(X))$ onto $ {L^ + }({C_0}(Y))$, then $ X$ and $ Y$ are homeomorphic. If $ X$ and $ Y$ are assumed to be compact then the same conclusion holds under the milder condition that $ \varphi $ is an affine isomorphism and $ \varphi ({I_{C(X)}})$ is order bounded.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1991-1037205-X
PII: S 0002-9939(1991)1037205-X
Article copyright: © Copyright 1991 American Mathematical Society



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