Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Projecting $ C(S)$ onto $ C\sb{0}(S)$


Author: H. Banilower
Journal: Proc. Amer. Math. Soc. 33 (1972), 349-354
MSC: Primary 46E10; Secondary 54C35
DOI: https://doi.org/10.1090/S0002-9939-1972-0295062-1
MathSciNet review: 0295062
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: If a locally compact Hausdorff space S has a denumerable discrete closed subspace N for which there exists a simultaneous extension E from $ C(N)$ into $ C(S)$ satisfying $ E({C_0}(N)) \subset {C_0}(S)$, then $ {C_0}(S)$ is uncomplemented in $ C(S)$. This holds whenever (i) S is not pseudocompact, or (ii) S is not countably compact and is a subspace of a basically disconnected space.


References [Enhancements On Off] (What's this?)

  • [B] W. G. Bade, Extensions of interpolation sets, Proc. Conference Functional Analysis (Irvine, Calif., 1966), Academic Press, London; Thompson, Washington, D.C., 1967, pp. 147-167. MR 36 #5693. MR 0222643 (36:5693)
  • [BCM] R. W. Bagley, E. H. Connell and J. D. McKnight, Jr., On properties characterizing pseudo-compact spaces, Proc. Amer. Math. Soc. 9 (1958), 500-506. MR 20 #3523. MR 0097043 (20:3523)
  • [C] J. B. Conway, Projections and retractions, Proc. Amer. Math. Soc. 17 (1966), 843-847. MR 33 #3253. MR 0195048 (33:3253)
  • [D] D. Dean, Subspaces of $ C(H)$ which are direct factors of $ C(H)$, Proc. Amer. Math. Soc. 16 (1965), 237-242. MR 30 #3352. MR 0173137 (30:3352)
  • [DS] N. Dunford and J. T. Schwartz, Linear operators. I: General theory, Pure and Appl. Math., vol. 7, Interscience, New York, 1958. MR 22 #8302. MR 0117523 (22:8302)
  • [FG$ _{1}$] N. J. Fine and L. Gillman, Extension of continuous functions in $ \beta N$, Bull. Amer. Math. Soc. 66 (1960), 376-381. MR 23 #A619. MR 0123291 (23:A619)
  • [FG$ _{2}$] -, Remote points in $ \beta R$, Proc. Amer. Math. Soc. 13 (1962), 29-36. MR 26 #732. MR 0143172 (26:732)
  • [GJ] L. Gillman and M. Jerison, Rings of continuous functions, The University Series in Higher Math., Van Nostrand, Princeton, N.J., 1960. MR 22 #6994. MR 0116199 (22:6994)
  • [P] R. S. Phillips, On linear transformations, Trans. Amer. Math. Soc. 48 (1940), 516-541. MR 2, 318. MR 0004094 (2:318c)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46E10, 54C35

Retrieve articles in all journals with MSC: 46E10, 54C35


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0295062-1
Keywords: Uncomplemented in $ C(S)$, simultaneous extension, denumerable discrete subspace, basically disconnected, extremally disconnected
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society