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A Note on Certain Subalgebras of C()

  • Anthony W. Hager (a1) and Donald G. Johnson (a2)

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Let be a completely regular Hausdorff space and C() the algebra of continuous real-valued functions on . In attempts to characterize abstractly those algebras that are isomorphic to C() for some , one produces subalgebras of C() which: (a) contain the constant functions, (b) separate points and closed sets in , (c) are closed under uniform convergence, and (d) are closed under inversion in C() (see, for example, (2; 5)).

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References

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1. Gillman, L. and Jerison, M., Rings of continuous functions (Princeton, 1960).
2. Henriksen, M. and Johnson, D. G., On the structure of a class of archimedean lattice-ordered algebras, Fund. Math., 80 (1961), 7394.
3. Henriksen, M., Johnson, D. G., and Isbell, J. R., Residue class fields of lattice-ordered algebras. Fund. Math., JO (1961), 107117.
4. Hewitt, E., Certain generalizations of the Weierstrass approximation theorem, Duke Math. J., 15 (1947), 410427.
5. Isbell, J. R., Algebras of uniformly continuous functions, Ann. of Math., 68 (1958), 96125.
6. Mrόwka, S. G., Some approximation theorems for rings of unbounded functions, Amer. Math. Soc. Not., 11 (1964), 666.
7. Mrόwka, S. G., ƒunctionals on uniformly closed rings of continuous functions, Fund. Math., Ifi (1958), 8187.
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