Prime ideals in algebras of continuous functions

Dales, H. G.; Loy, R. J.

doi:10.1090/S0002-9939-1986-0857934-6

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
DOI: https://doi.org/10.1090/S0002-9939-1986-0857934-6
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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