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Transactions of the American Mathematical Society

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Uncomplemented $ C(X)$-subalgebras of $ C(X)$


Author: John Warren Baker
Journal: Trans. Amer. Math. Soc. 186 (1973), 1-15
MSC: Primary 46E15
DOI: https://doi.org/10.1090/S0002-9947-1973-0331034-1
MathSciNet review: 0331034
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Abstract: In this paper, the uncomplemented subalgebras of the Banach algebra $ C(X)$ which are isometrically and algebraically isomorphic to $ C(X)$ are investigated. In particular, it is shown that if X is a 0-dimensional compact metric space with its $ \omega $th topological derivative $ {X^{(\omega )}}$ nonempty, then there is an uncomplemented subalgebra of $ C(X)$ isometrically and algebraically isomorphic to $ C(X)$.

For each ordinal $ \alpha \geq 1$, a class $ {\mathcal{C}_\alpha }$ of homeomorphic 0-dimensional uncountable compact metric spaces is introduced. It is shown that each uncountable 0-dimensional compact metric space contains an open-and-closed subset which belongs to some $ {\mathcal{C}_\alpha }$.


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  • [1] D. Amir, Continuous function spaces with the separable projection property, Bull. Res. Council of Israel Sect. F 10F (1962), 163-164. MR 27 #566. MR 0150570 (27:566)
  • [2] -, Projections onto continuous function spaces, Proc. Amer. Math. Soc. 15 (1964), 396-402. MR 29 #2634. MR 0165350 (29:2634)
  • [3] R. Arens, Projections on continuous function spaces, Duke Math. J. 32 (1965), 469-478. MR 31 #6108. MR 0181882 (31:6108)
  • [4] J. W. Baker, Some uncomplemented subspaces of $ C(X)$ of the type $ C(Y)$, Studia Math. 36 (1970), 85-103. MR 43 #1113. MR 0275356 (43:1113)
  • [5] -, Compact spaces hoemomorphic to a ray of ordinals, Fund. Math. 76 (1972), 19-27. MR 0307197 (46:6317)
  • [6] -, Dispersed images of topological spaces and uncomplemented subspaces of $ C(X)$, Proc. Amer. Math. Soc. 41 (1973), 309-314. MR 0320984 (47:9517)
  • [7] -, Ordinal subspaces of topological spaces, General Topology and Appl. 3 (1973), 85-91. MR 0324623 (48:2973)
  • [8] -, Projection constants for $ C(S)$ spaces with the separable projection property, Proc. Amer. Math. Soc. 41 (1973), 201-204. MR 0320707 (47:9242)
  • [9] J. Baker and R. Lacher, Some mappings which do not admit an averaging operator (to appear). MR 0511834 (58:23527)
  • [10] N. Bourbaki, Eléments de mathématique. Part. 1. Les structures fondamentales de l'analyse. Livre III: Topologie générale, Actualités Sci. Indust., no. 1029, Hermann, Paris, 1947; English transl., Hermann, Paris; Addison-Wesley, Reading, Mass., 1966. MR 9, 261; 34 #5044b.
  • [11] S. Ditor, On a lemma of Milutin concerning averaging operators in continuous function spaces, Trans. Amer. Math. Soc. 149 (1970), 443-452. MR 0435921 (55:8872)
  • [12] -, Averaging operators in $ C(S)$ and lower semicontinuous sections of continuous maps, Trans. Amer. Math. Soc. 175 (1973), 195-208. MR 0312228 (47:790)
  • [13] J. Dugundji, Topology, Allyn and Bacon, Boston, Mass., 1966. MR 33 #1824. MR 0193606 (33:1824)
  • [14] 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)
  • [15] R. Engelking, Outline of general topology, PWN, Warsaw, 1965; English transl., North-Holland, Amsterdam; Interscience, New York, 1968. MR 36 #4508; 37 #5836. MR 0230273 (37:5836)
  • [16] F. Hausdorff, Set theory, 2nd ed., Chelsea, New York, 1957. MR 0086020 (19:111a)
  • [17] E. Hewitt and K. Stromberg, Real and abstract analysis. A modern treatment of the theory of functions of a real variable, Springer-Verlag, New York, 1965, MR 32 #5826. MR 0188387 (32:5826)
  • [18] J. Kelley, General topology, Van Nostrand, Princeton, N. J., 1955. MR 16, 1136. MR 0070144 (16:1136c)
  • [19] K. Morita and S. Hanai, Closed mappings and metric spaces, Proc. Japan Acad. 32 (1956), 10-14. MR 19, 299. MR 0087077 (19:299a)
  • [20] A. Pełczyński and Z. Semadeni, Spaces of continuous functions. III. Spaces $ C(\Omega )$ for $ \Omega $ without perfect subsets, Studia Math. 18 (1959), 211-222. MR 21 #6528. MR 0107806 (21:6528)
  • [21] A. Pełczyński, Linear extensions, linear averagings, and their applications to linear topological classification of spaces of continuous functions, Rozprawy Mat. 58 (1968), 92 pp. MR 37 #3335. MR 0227751 (37:3335)
  • [22] -, On $ C(S)$-subspaces of separable Banach spaces, Studia Math. 31 (1968), 513-522. MR 38 #2578. MR 0234261 (38:2578)
  • [23] R. S. Pierce, Existence and uniqueness theorems for extensions of zero-dimensional compact metric spaces, Trans. Amer. Math. Soc. 148 (1970), 1-21. MR 40 #8011. MR 0254804 (40:8011)
  • [24] Z. Semadeni, Sur les ensembles clairsemés, Rozprawy Mat. 19 (1959), 1-39. MR 21 #6571. MR 0107849 (21:6571)
  • [25] W. Sierpinski, General topology, 2nd ed., Univ. of Toronto Press, Toronto, 1956. MR 0050870 (14:394f)
  • [26] A. H. Stone, Metrizability of decomposition spaces, Proc. Amer. Math. Soc. 7 (1956), 690-700. MR 19, 299. MR 0087078 (19:299b)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1973-0331034-1
Keywords: Banach spaces of continuous functions, Banach algebras of continuous functions, complemented subspaces of $ C(X)$, averaging operators, compact 0-dimensional metric spaces, Boolean algebras
Article copyright: © Copyright 1973 American Mathematical Society

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