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Prime ideals in algebras of continuous functions

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<mark>Journal publication date</mark>1986
<mark>Journal</mark>Proceedings of the American Mathematical Society
Issue number3
Number of pages5
Pages (from-to)426-430
Publication StatusPublished
<mark>Original language</mark>English


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.