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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Prime ideals in algebras of continuous functions

Authors: H. G. Dales and R. J. Loy
Journal: Proc. Amer. Math. Soc. 98 (1986), 426-430
MSC: Primary 46J10; Secondary 46J20, 54C40
MathSciNet review: 857934
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Abstract: Let $ {X_0}$ be a compact Hausdorff space, and let $ {\mathbf{C}}({X_0})$ be the Banach algebra of all continuous complex-valued functions on $ {X_0}$. It is known that, assuming the continuum hypothesis, any nonmaximal, prime ideal $ {\mathbf{P}}$ such that $ \vert{\mathbf{C}}({X_0})/{\mathbf{P}}\vert = {2^{{\aleph _0}}}$ is the kernel of a discontinuous homomorphism from $ {\mathbf{C}}({X_0})$ into some Banach algebra. Here we consider the converse question of which ideals can be the kernels of such a homomorphism. Partial results are obtained in the case where $ {X_0}$ is metrizable.

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PII: S 0002-9939(1986)0857934-6
Keywords: Prime ideals, discontinuous homomorphisms
Article copyright: © Copyright 1986 American Mathematical Society