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

HTML articles powered by AMS MathViewer

- by John Warren Baker PDF
- Trans. Amer. Math. Soc.
**186**(1973), 1-15 Request permission

## 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 ${\mathcal {C}_\alpha }$.

## References

- D. Amir,
*Continuous functions’ spaces with the separable projection property*, Bull. Res. Council Israel Sect. F**10F**(1962), 163–164 (1962). MR**150570** - 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 - 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 - John Warren Baker,
*Compact spaces homeomorphic to a ray of ordinals*, Fund. Math.**76**(1972), no. 1, 19–27. MR**307197**, DOI 10.4064/fm-76-1-19-27 - John Warren Baker,
*Dispersed images of topological spaces and uncomplemented subspaces of $C(X)$*, Proc. Amer. Math. Soc.**41**(1973), 309–314. MR**320984**, DOI 10.1090/S0002-9939-1973-0320984-3 - John Warren Baker,
*Ordinal subspaces of topological spaces*, General Topology and Appl.**3**(1973), 85–91. MR**324623** - John Warren Baker,
*Projection constants for $C(S)$ spaces with the separable projection property*, Proc. Amer. Math. Soc.**41**(1973), 201–204. MR**320707**, DOI 10.1090/S0002-9939-1973-0320707-8 - John Warren Baker and R. C. Lacher,
*Some mappings which do not admit an averaging operator*, Pacific J. Math.**62**(1976), no. 1, 43–48. MR**511834**
N. Bourbaki, - Seymour Z. Ditor,
*On a lemma of Milutin concerning averaging operators in continuous function spaces*, Trans. Amer. Math. Soc.**149**(1970), 443–452. MR**435921**, DOI 10.1090/S0002-9947-1970-0435921-2 - 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 - James Dugundji,
*Topology*, Allyn and Bacon, Inc., Boston, Mass., 1966. MR**0193606** - 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** - R. Engelking,
*Outline of general topology*, North-Holland Publishing Co., Amsterdam; PWN—Polish Scientific Publishers, Warsaw; Interscience Publishers Division John Wiley & Sons, Inc., New York, 1968. Translated from the Polish by K. Sieklucki. MR**0230273** - Felix Hausdorff,
*Set theory*, Chelsea Publishing Co., New York, 1957. Translated by John R. Aumann, et al. MR**0086020** - Edwin Hewitt and Karl Stromberg,
*Real and abstract analysis. A modern treatment of the theory of functions of a real variable*, Springer-Verlag, New York, 1965. MR**0188387** - John L. Kelley,
*General topology*, D. Van Nostrand Co., Inc., Toronto-New York-London, 1955. MR**0070144** - Kiiti Morita and Sitiro Hanai,
*Closed mappings and metric spaces*, Proc. Japan Acad.**32**(1956), 10–14. MR**87077** - 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**107806**, DOI 10.4064/sm-18-2-211-222 - 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** - A. Pełczyński,
*On $C(S)$-subspaces of separable Banach spaces*, Studia Math.**31**(1968), 513–522. MR**234261**, DOI 10.4064/sm-31-5-513-522 - R. S. Pierce,
*Existence and uniqueness theorems for extensions of zero-dimensional compact metric spaces*, Trans. Amer. Math. Soc.**148**(1970), 1–21. MR**254804**, DOI 10.1090/S0002-9947-1970-0254804-4 - Z. Semadeni,
*Sur les ensembles clairsemés*, Rozprawy Mat.**19**(1959), 39 pp. (1959) (French). MR**107849** - Waclaw Sierpinski,
*General topology*, Mathematical Expositions, No. 7, University of Toronto Press, Toronto, 1952. Translated by C. Cecilia Krieger. MR**0050870** - A. H. Stone,
*Metrizability of decomposition spaces*, Proc. Amer. Math. Soc.**7**(1956), 690–700. MR**87078**, DOI 10.1090/S0002-9939-1956-0087078-6

*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.

## Additional Information

- © Copyright 1973 American Mathematical Society
- 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