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 is an algebra of real-valued functions on a set equipped with the weak topology for , and if separates its zero sets, then many results valued for equipped with a strict topology remain true when is equipped with a strict topology. The concepts of -additivity and tight positive linear operators are introduced. It is shown that if is a positive linear map on into -separating algebra and if , then there exists a continuous function on (the domain of elements in ) into such that if and only if is an algebraic homomorphism and -additive.

**[1]**F. F. Bonsall, J. Lindenstrauss, and R. R. Phelps,*Extreme positive operators on algebras of functions*, Math. Scand.**18**(1966), 161–182. MR**0209863****[2]**A. J. Ellis,*Extreme positive operators*, Quart. J. Math. Oxford Ser. (2)**15**(1964), 342–344. MR**0173950****[3]**J. Hoffmann-Jørgensen,*A generalization of the strict topology*, Math. Scand.**30**(1972), 313–323. MR**0318857****[4]**R. B. Kirk,*Algebras of bounded real-valued functions. I*, Nederl. Akad. Wetensch. Proc. Ser. A 75=Indag. Math.**34**(1972), 443–451. MR**0320725****[5]**R. B. Kirk and J. A. Crenshaw,*A generalized topological measure theory*(to appear).**[6]**J. D. Knowles,*Measures on topological spaces*, Proc. London Math. Soc. (3)**17**(1967), 139–156. MR**0204602****[7]**S. E. Mosiman,*Strict topologies and the ordered vector space*(to appear).**[8]**Steven E. Mosiman and Robert F. Wheeler,*The strict topology in a completely regular setting: relations to topological measure theory*, Canad. J. Math.**24**(1972), 873–890. MR**0328567****[9]**F. Dennis Sentilles,*Bounded continuous functions on a completely regular space*, Trans. Amer. Math. Soc.**168**(1972), 311–336. MR**0295065**, 10.1090/S0002-9947-1972-0295065-1**[10]**V. S. Varadarajan,*Measures on topological spaces*, Mat. Sb. (N.S.)**55 (97)**(1961), 35–100 (Russian). MR**0148838**

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DOI:
https://doi.org/10.1090/S0002-9939-1974-0370150-1

Keywords:
Algebras of functions,
strict topologies,
positive linear operator,
-additive,
-additive,
tight,
extreme linear operator,
algebraic homomorphism

Article copyright:
© Copyright 1974
American Mathematical Society