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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

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


Author: John Warren Baker
Journal: Trans. Amer. Math. Soc. 186 (1973), 1-15
MSC: Primary 46E15
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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DOI: http://dx.doi.org/10.1090/S0002-9947-1973-0331034-1
PII: S 0002-9947(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