Injective Banach spaces of continuous functions
HTML articles powered by AMS MathViewer
- by John Wolfe PDF
- Trans. Amer. Math. Soc. 235 (1978), 115-139 Request permission
Abstract:
A description is given of the compact Hausdorff spaces S such that the Banach space $C(S)$ of continuous functions on S is a ${P_\lambda }$-space for $\lambda < 3$ (under the assumption that S satisfies the countable chain condition). The existence of extension operators from $C({X^\ast }\backslash X)$ to $C({X^\ast })$ is examined under the assumption that $C({X^\ast })$ is injective where ${X^\ast }$ is some compactification of a locally compact extremally disconnected Hausdorff space X (if $C(S)$ is injective, S is of this form). Some new examples of injective spaces $C(S)$ are given.References
- D. Amir, Continuous functions’ spaces with the bounded extension property, Bull. Res. Council Israel Sect. F 10F (1962), 133–138 (1962). MR 143026
- D. Amir, Projections onto continuous function spaces, Proc. Amer. Math. Soc. 15 (1964), 396–402. MR 165350, DOI 10.1090/S0002-9939-1964-0165350-3
- Dan Amir, On projections and simultaneous extensions, Israel J. Math. 2 (1964), 245–248. MR 180845, DOI 10.1007/BF02759740 —, Continuous function spaces with small projection constants, Proc. Sympos. on Functional Analysis, Hiroshima Univ., 1965.
- Richard Arens, Projections on continuous function spaces, Duke Math. J. 32 (1965), 469–478. MR 181882
- John Warren Baker, Some uncomplemented subspaces of $C(X)$ of the type $C(Y)$, Studia Math. 36 (1970), 85–103. MR 275356, DOI 10.4064/sm-36-2-85-103 H. Banilower, Simultaneous extensions and projections in $C(S)$, Thesis.
- Y. Benyamini, Constants of simultaneous extension of continuous functions, Israel J. Math. 16 (1973), 258–262. MR 341049, DOI 10.1007/BF02756705
- Henry B. Cohen, Injective envelopes of Banach spaces, Bull. Amer. Math. Soc. 70 (1964), 723–726. MR 184060, DOI 10.1090/S0002-9904-1964-11189-7
- H. B. Cohen, M. A. Labbe, and J. Wolfe, Norm reduction of averaging operators, Proc. Amer. Math. Soc. 35 (1972), 519–523. MR 324389, DOI 10.1090/S0002-9939-1972-0324389-X
- John B. Conway, Projections and retractions, Proc. Amer. Math. Soc. 17 (1966), 843–847. MR 195048, DOI 10.1090/S0002-9939-1966-0195048-9
- Mahlon M. Day, Normed linear spaces, Reihe: Reelle Funktionen, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1958. MR 0094675
- David W. Dean, Subspaces of $C(H)$ which are direct factors of $C(H)$, Proc. Amer. Math. Soc. 16 (1965), 237–242. MR 173137, DOI 10.1090/S0002-9939-1965-0173137-1 S. Z. Ditor, Linear operators of averaging and extension, Thesis, Univ. of California, Berkeley, 1968.
- Seymour Z. Ditor, Averaging operators in $C(S)$ and lower semicontinuous sections of continuous maps, Trans. Amer. Math. Soc. 175 (1973), 195–208. MR 312228, DOI 10.1090/S0002-9947-1973-0312228-8
- 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
- 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
- Andrew M. Gleason, Projective topological spaces, Illinois J. Math. 2 (1958), 482–489. MR 121775
- Dwight B. Goodner, Projections in normed linear spaces, Trans. Amer. Math. Soc. 69 (1950), 89–108. MR 37465, DOI 10.1090/S0002-9947-1950-0037465-6
- Morisuke Hasumi, The extension property of complex Banach spaces, Tohoku Math. J. (2) 10 (1958), 135–142. MR 100781, DOI 10.2748/tmj/1178244708
- J. R. Isbell and Z. Semadeni, Projection constants and spaces of continuous functions, Trans. Amer. Math. Soc. 107 (1963), 38–48. MR 146649, DOI 10.1090/S0002-9947-1963-0146649-7
- R. Kaufman, A type of extension of Banbach spaces, Acta Sci. Math. (Szeged) 27 (1966), 163–166. MR 205037
- J. L. Kelley, Banach spaces with the extension property, Trans. Amer. Math. Soc. 72 (1952), 323–326. MR 45940, DOI 10.1090/S0002-9947-1952-0045940-5
- John L. Kelley, General topology, D. Van Nostrand Co., Inc., Toronto-New York-London, 1955. MR 0070144
- H. Elton Lacey, The isometric theory of classical Banach spaces, Die Grundlehren der mathematischen Wissenschaften, Band 208, Springer-Verlag, New York-Heidelberg, 1974. MR 0493279
- H. Elton Lacey and H. B. Cohen, On injective envelopes of Banach spaces, J. Functional Analysis 4 (1969), 11–30. MR 0243322, DOI 10.1016/0022-1236(69)90019-6
- Joram Lindenstrauss, Extension of compact operators, Mem. Amer. Math. Soc. 48 (1964), 112. MR 179580
- Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces, Lecture Notes in Mathematics, Vol. 338, Springer-Verlag, Berlin-New York, 1973. MR 0415253
- Leopoldo Nachbin, A theorem of the Hahn-Banach type for linear transformations, Trans. Amer. Math. Soc. 68 (1950), 28–46. MR 32932, DOI 10.1090/S0002-9947-1950-0032932-3
- A. Pełczyński, Linear extensions, linear averagings, and their applications to linear topological classification of spaces of continuous functions, Dissertationes Math. (Rozprawy Mat.) 58 (1968), 92. MR 227751
- R. S. Phillips, On linear transformations, Trans. Amer. Math. Soc. 48 (1940), 516–541. MR 4094, DOI 10.1090/S0002-9947-1940-0004094-3
- Baltasar Rodríguez-Salinas Palero, Some problems and theorems on the extension of linear mappings, Rev. Acad. Ci. Madrid 65 (1971), 677–704 (Spanish). MR 305030
- Haskell P. Rosenthal, On injective Banach spaces and the spaces $L^{\infty }(\mu )$ for finite measure $\mu$, Acta Math. 124 (1970), 205–248. MR 257721, DOI 10.1007/BF02394572
- 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
- Zbigniew Semadeni, Banach spaces of continuous functions. Vol. I, Monografie Matematyczne, Tom 55, PWN—Polish Scientific Publishers, Warsaw, 1971. MR 0296671 R. J. Whitley, Some ${P_\lambda }$ Banach spaces, Conf. on Projections, Clemson, South Carolina, 1968. J. Wolfe, Injective Banach spaces of type $C(T)$, Thesis, Univ. of California, Berkeley, 1971.
- John Wolfe, Injective Banach spaces of type $C(T)$, Israel J. Math. 18 (1974), 133–140. MR 352959, DOI 10.1007/BF02756867
Additional Information
- © Copyright 1978 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 235 (1978), 115-139
- MSC: Primary 46E15; Secondary 46M10
- DOI: https://doi.org/10.1090/S0002-9947-1978-0461113-4
- MathSciNet review: 0461113