On dominated extensions in function algebras

Globevnik, J.

doi:10.1090/S0002-9939-1980-0572304-0

On dominated extensions in function algebras
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by J. Globevnik

Proc. Amer. Math. Soc. 79 (1980), 571-576

DOI: https://doi.org/10.1090/S0002-9939-1980-0572304-0

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Abstract:

The Bishop-Gamelin interpolation theorem asserts that given a compact Hausdorff space K, a closed subspace A of $C(K)$, a positive continuous function p on K and a closed set $F \subset K$ such that every measure in the annihilator of A vanishes on F, every function $f \in C(F)$ satisfying $|f(s)| \leqslant p(s)(s \in F)$ extends to a function $\tilde f \in A$ satisfying $|\tilde f(z)| \leqslant p(z)(z \in K)$. In the paper we consider a special case where the theorem is extended to the situation when the dominating function is nonnegative.

References

E. M. Alfsen and B. Hirsberg, On dominated extensions in linear subspaces of ${\cal C}_{C}\,(X)$, Pacific J. Math. 36 (1971), 567–584. MR 283553
Errett Bishop, A general Rudin-Carleson theorem, Proc. Amer. Math. Soc. 13 (1962), 140–143. MR 133462, DOI 10.1090/S0002-9939-1962-0133462-4
T. W. Gamelin, Restrictions of subspaces of $C(X)$, Trans. Amer. Math. Soc. 112 (1964), 278–286. MR 162132, DOI 10.1090/S0002-9947-1964-0162132-8
Theodore W. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1969. MR 0410387
J. Globevnik, Analytic extensions and selections, Trans. Amer. Math. Soc. 254 (1979), 171–177. MR 539913, DOI 10.1090/S0002-9947-1979-0539913-2
Per Hag, A generalization of the Rudin-Carleson theorem, Proc. Amer. Math. Soc. 43 (1974), 341–344. MR 338755, DOI 10.1090/S0002-9939-1974-0338755-1
Per Hag, Erratum to: “A generalization of the Hudin-Carleson theorem” (Proc. Amer. Math. Soc. 43 (1974), 341–344), Proc. Amer. Math. Soc. 60 (1976), 377. MR 415325, DOI 10.1090/S0002-9939-1976-0415325-X
Per Hag, Restrictions of convex subsets of $C(X)$, Trans. Amer. Math. Soc. 233 (1977), 283–294. MR 467264, DOI 10.1090/S0002-9947-1977-0467264-1
Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0133008
Walter Roth, A general Rudin-Carlson theorem in Banach-spaces, Pacific J. Math. 73 (1977), no. 1, 197–213. MR 467259

Simultaneous extensions and projections in spaces of continuous functions

Edgar Lee Stout, The theory of uniform algebras, Bogden & Quigley, Inc., Publishers, Tarrytown-on-Hudson, N.Y., 1971. MR 0423083