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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Uncomplemented $C(X)$-subalgebras of $C(X)$
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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 ${\mathcal {C}_\alpha }$.
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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