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Proceedings of the American Mathematical Society

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



Positive linear operators continuous for strict topologies

Author: J. A. Crenshaw
Journal: Proc. Amer. Math. Soc. 46 (1974), 79-85
MSC: Primary 46E10
MathSciNet review: 0370150
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Abstract: If $ A$ is an $ SW$ algebra of real-valued functions on a set $ X$ equipped with the weak topology for $ A$, and if $ A$ separates its zero sets, then many results valued for $ {C^b}(X)$ equipped with a strict topology remain true when $ A$ is equipped with a strict topology. The concepts of $ \alpha $-additivity and tight positive linear operators are introduced. It is shown that if $ T$ is a positive linear map on $ A$ into $ z$-separating $ SW$ algebra $ B$ and if $ T({1_A}) = {1_B}$, then there exists a continuous function $ \phi $ on $ Y$ (the domain of elements in $ B$) into $ X$ such that $ Tf(y) = f(\phi (y))$ if and only if $ T$ is an algebraic homomorphism and $ \tau $-additive.

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Keywords: Algebras of functions, strict topologies, positive linear operator, $ \sigma $-additive, $ \tau $-additive, tight, extreme linear operator, algebraic homomorphism
Article copyright: © Copyright 1974 American Mathematical Society

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