Simultaneous extensions from discrete subspaces

Banilower, H.

doi:10.1090/S0002-9939-1972-0319138-5

Simultaneous extensions from discrete subspaces
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Proc. Amer. Math. Soc. 36 (1972), 451-455 Request permission

Abstract:

If N is a denumerable, discrete, normally embedded subspace of the completely regular space S, then any bounded linear operator from $C(N)$ into $C(S)$ that extends functions in ${C_0}(N)$ necessarily extends all bounded functions on some infinite subset of N (that are zero elsewhere on N). For compact S, such operators exist whenever $C(S)$ contains a subspace isometric to (m). It is also shown, assuming the continuum hypothesis, that if S is a locally compact F-space and ${C_0}(S)$ is complemented in $C(S)$, then S is countably compact.

References

H. Banilower, Projecting $C(S)$ onto $C_{0}(S)$, Proc. Amer. Math. Soc. 33 (1972), 349–354. MR 295062, DOI 10.1090/S0002-9939-1972-0295062-1
W. W. Comfort, Retractions and other continuous maps from $\beta X$ onto $\beta X\sbs X$, Trans. Amer. Math. Soc. 114 (1965), 1–9. MR 185571, DOI 10.1090/S0002-9947-1965-0185571-9
Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
Dwight B. Goodner, Subspaces of $C(S)$ isometric to $m$, J. London Math. Soc. (2) 3 (1971), 488–492. MR 287287, DOI 10.1112/jlms/s2-3.3.488
Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0116199
W. Holsztyński, Continuous mappings induced by isometries of spaces of continuous function, Studia Math. 26 (1966), 133–136. MR 193491, DOI 10.4064/sm-26-2-133-136
Joram Lindenstrauss, On the extension of operators with range in a $C(K)$ space, Proc. Amer. Math. Soc. 15 (1964), 218–225. MR 167822, DOI 10.1090/S0002-9939-1964-0167822-4
Haskell P. Rosenthal, On relatively disjoint families of measures, with some applications to Banach space theory, Studia Math. 37 (1970), 13–36. MR 270122, DOI 10.4064/sm-37-1-13-36