A characterization of -separating algebras

Author:
Shankar Hegde

Journal:
Proc. Amer. Math. Soc. **73** (1979), 40-44

MSC:
Primary 46J10; Secondary 54C50

DOI:
https://doi.org/10.1090/S0002-9939-1979-0512055-3

MathSciNet review:
512055

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Abstract | References | Similar Articles | Additional Information

Abstract: Let *A* be a uniformly closed point separating algebra of bounded real valued functions on a set *X*, containing the constant functions. *A* is called *z*-separating if whenever are disjoint zero sets of members of *A* there is some with and . We prove that *A* is *z*-separating if and only if *A* consists of precisely those bounded real valued functions *f* on *X* for which is a zero set of some member of *A* for every closed set *C* of real line.

**[1]**R. M. Brooks,*On Wallman compactifications*, Fund. Math.**60**(1967), 157-173. MR**0210069 (35:964)****[2]**N. Dunford and J. T. Schwartz,*Linear operators*Part I, Interscience, New York, 1958.**[3]**O. Frink,*Compactifications and seminormal spaces*, Amer. J. Math.**86**(1964), 602-607. MR**0166755 (29:4028)****[4]**Z. Frolik,*Three uniformities associated with uniformly continuous functions*, Symposia Math.**17**(1976), 69-80. MR**0478110 (57:17599)****[5]**A. W. Hager,*On inverse closed subalgebras of*, Proc. London Math. Soc.**19**(1969), 233-257. MR**0244948 (39:6261)****[6]**-,*Real-valued functions on Alexandroff*(*zero-set*)*spaces*, Comment. Math. Univ. Carolinae**16**(1975), 755-769. MR**0394547 (52:15348)****[7]**-,*A class of function algebras*(*and compactifications and uniform spaces*), Symposia Math.**17**(1976), 11-23. MR**0425891 (54:13841)****[8]**R. B. Kirk,*Algebras of bounded real valued functions*. I, II, Nederl. Akad. Wetensh. Proc. Ser. A 75 = Indag. Math.**34**(1972), 443-463. MR**0320725 (47:9260a)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1979-0512055-3

Keywords:
*z*-separating algebra,
Wallman compact space,
-continuity and -uniform continuity

Article copyright:
© Copyright 1979
American Mathematical Society