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

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- by John Warren Baker
- Trans. Amer. Math. Soc.
**186**(1973), 1-15 - DOI: https://doi.org/10.1090/S0002-9947-1973-0331034-1
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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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## Bibliographic 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