𝐶(𝛼) preserving operators on 𝐶(𝐾) spaces

Wolfe, John

doi:10.1090/S0002-9947-1982-0667169-9

$C(\alpha )$ preserving operators on spaces

Author: John Wolfe
Journal: Trans. Amer. Math. Soc. 273 (1982), 705-719
MSC: Primary 46E15; Secondary 46B25
DOI: https://doi.org/10.1090/S0002-9947-1982-0667169-9
MathSciNet review: 667169
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Abstract: Let $A:C(K) \to X$ be a bounded linear operator where is a compact Hausdorff space and is a separable Banach space. Sufficient conditions are given for to be an isomorphism (into) when restricted to a subspace of , such that is isometrically isomorphic to a space $C(\alpha )$ of continuous functions on the space of ordinal numbers less than or equal to the countable ordinal $\alpha$ .

[1] Dale E. Alspach, Quotients of 𝐶[0,1] with separable dual, Israel J. Math. 29 (1978), no. 4, 361–384. MR 491925, https://doi.org/10.1007/BF02761174
[2] -, norming subsets of $C{[0,1]^\ast}$ , Studia Math. (to appear).
[3] D. E. Alspach and Y. Benyamini, quotients of separable ${L_\infty }$ spaces, Israel J. Math. (to appear).
[4] John Warren Baker, Compact spaces homeomorphic to a ray of ordinals, Fund. Math. 76 (1972), no. 1, 19–27. MR 307197, https://doi.org/10.4064/fm-76-1-19-27
[5] Y. Benyamini, An extension theorem for separable Banach spaces, Israel J. Math. 29 (1978), no. 1, 24–30. MR 482075, https://doi.org/10.1007/BF02760399
[6] C. Bessaga and A. Pełczyński, Spaces of continuous functions. IV. On isomorphical classification of spaces of continuous functions, Studia Math. 19 (1960), 53–62. MR 113132, https://doi.org/10.4064/sm-19-1-53-62
[7] J. Diestel and J. J. Uhl Jr., Vector measures, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis; Mathematical Surveys, No. 15. MR 0453964
[8] Felix Hausdorff, Set theory, Second edition. Translated from the German by John R. Aumann et al, Chelsea Publishing Co., New York, 1962. MR 0141601
[9] K. Kuratowski, Topology. Vol. II, New edition, revised and augmented. Translated from the French by A. Kirkor, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe Polish Scientific Publishers, Warsaw, 1968. MR 0259835
[10] S. Mazurbiewicz and W. Sierpinski, Contributions à la topologie des ensembles dinombrables, Fund. Math. 1 (1920), 17-27.
[11] A. Pełczyński, On 𝐶(𝑆)-subspaces of separable Banach spaces, Studia Math. 31 (1968), 513–522. MR 234261, https://doi.org/10.4064/sm-31-5-513-522
[12] 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 0227751
[13] Haskell P. Rosenthal, On relatively disjoint families of measures, with some applications to Banach space theory, Studia Math. 37 (1970), 13–36. MR 270122, https://doi.org/10.4064/sm-37-1-13-36
[14] Haskell P. Rosenthal, On factors of 𝐶([0,1]) with non-separable dual, Israel J. Math. 13 (1972), 361–378 (1973); correction, ibid. 21 (1975), no. 1, 93–94. MR 388063, https://doi.org/10.1007/BF02762811
[15] Wacław Sierpiński, Cardinal and ordinal numbers, Second revised edition. Monografie Matematyczne, Vol. 34, Państowe Wydawnictwo Naukowe, Warsaw, 1965. MR 0194339
[16] W. Szlenk, The non-existence of a separable reflexive Banach space universal for all separable reflexive Banach spaces, Studia Math. 30 (1968), 53–61. MR 227743, https://doi.org/10.4064/sm-30-1-53-61
[17] D. Alspach and Y. Benyamini, Primariness of spaces of continuous functions on ordinals, Israel J. Math. 27 (1977), no. 1, 64–92. MR 440349, https://doi.org/10.1007/BF02761606
[18] Pierre Billard, Sur la primarité des espaces 𝐶(𝛼), Studia Math. 62 (1978), no. 2, 143–162 (French). MR 499961, https://doi.org/10.4064/sm-62-2-143-162