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F-Rings of Continuous Functions I

Published online by Cambridge University Press:  20 November 2018

Barron Brainerd
Barron Brainerd*
Affiliation:
The University of Western Ontarioand, Summer Research Institute of the Canadian Mathematical Congress
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It is well known (2, 4) that the ring of all real (complex) continuous functions on a compact Hausdorff space can be characterized algebraically as a Banach algebra which satisfies certain additional intrinsic conditions. It might be expected that rings of all continuous functions on other topological spaces also have algebraic characterizations. The main purpose of this note is to discuss two such characterizations. In both cases the characterizations are given in the terms of the theory of F-brings (1). In one case a characterization is given for the ring of all (real) continuous functions on a generalized P-space, that is, a zero-dimensional topological space in which the class of open-closed sets forms a σ-algebra. A Hausdorff generalized P-space is a P-space in the terminology of (3). In the other case a theorem of Sikorski (6) is employed to give a characterization of the ring of all (real) continuous functions on an upper X1-compact P-space.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 11 , 1959 , pp. 80 - 86
Copyright
Copyright © Canadian Mathematical Society 1959

References

1. Brainerd, B., On a class of lattice-ordered rings, Proc. Amer. Math. Soc, 8 (1957), 673–83.Google Scholar
2. Gelfand, I., Normierte Ringe, Rec Math. (Mat. Sbornik) N.S., 9, 51 (1941), 3–24.Google Scholar
3. Gillman, L. and Henriksen, M., Concerning rings of continuous functions, Trans. Amer. Math. Soc, 77 (1954), 340-62.Google Scholar
4. Hewitt, E., Rings of real-valued continuous functions I, Trans. Amer. Math. Soc, 64 (1948), 45–99.Google Scholar
5. Morrison, D. R., Bi-regular rings and the ideal lattice isomorphisms, Proc. Amer. Math. Soc, 16 (1955), 46–9.Google Scholar
6. Sikorski, R., Remarks on some topological spaces of high power, Fund. Math., 37 (1950), 125–36.Google Scholar