Closed ideals in the functionally countable subalgebra of C(X)
Keywords:zero-dimensional space, functionally countable subalgebra, m-topology, closed ideal, ec-filter, ec-ideal, P-space
In this paper, closed ideals in Cc(X), the functionally countable subalgebra of C(X), with the mc-topology, is studied. We show that if
X is CUC-space, then C*c(X) with the uniform norm-topology is a Banach algebra. Closed ideals in Cc(X) as a modified countable analogue of closed ideals in C(X) with the m-topology are characterized. For a zero-dimensional space X, we show that a proper ideal in Cc(X) is closed if and only if it is an intersection of maximal ideals of Cc(X). It is also shown that every ideal in Cc(X) with the mc-topology is closed if and only if X is a P-space if and only if every ideal in C(X) with the m-topology is closed. Moreover, for a strongly zero-dimensional space X, it is proved that a properly closed ideal in C*c(X) is an intersection of maximal ideals of C*c(X) if and only if X is pseudo compact. Finally, we show that if X is a P-space, then the family of ec-ultrafilters and zc-ultrafilter coincide.
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