Positive linear operators continuous for strict topologies

Author:
J. A. Crenshaw

Journal:
Proc. Amer. Math. Soc. **46** (1974), 79-85

MSC:
Primary 46E10

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

MathSciNet review:
0370150

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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**35**#759. MR**0209863 (35:759)****[2]**A. J. Ellis,*Extreme positive operators*, Quart. J. Math. Oxford Ser. (2)**15**(1964), 342-344. MR**30**#4157. MR**0173950 (30:4157)****[3]**J. Hoffman-Jørgensen,*A generalization of the strict topology*, Math. Scand.**30**(1972), 313-323. MR**0318857 (47:7403)****[4]**R. B. Kirk,*Algebras of bounded real-valued functions*. I, II, Nederl. Akad. Wetensch. Proc. Ser. A**75**= Indag. Math.**34**(1972), 443-463. MR**0320725 (47:9260a)****[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**34**#4441. MR**0204602 (34:4441)****[7]**S. E. Mosiman,*Strict topologies and the ordered vector space*(to appear).**[8]**S. E. Mosiman and R. F. Wheeler,*The strict topology in a completely regular setting: relations to topological measure theory*, Canad. J. Math.**24**(1972), 873-890. MR**0328567 (48:6909)****[9]**F. D. Sentilles,*Bounded continuous functions on a completely regular space*, Trans. Amer. Math. Soc.**168**(1972), 311-336. MR**45**#4133. MR**0295065 (45:4133)****[10]**V. S. Varadarajan,*Measures on topological spaces*, Mat. Sb.**55**(97) (1961), 35-100; English transl., Amer. Math. Soc. Transl. (2)**48**(1965), 161-228. MR**26**#6342. MR**0148838 (26:6342)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
46E10

Retrieve articles in all journals with MSC: 46E10

Additional Information

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