TY - JOUR
AU - Veisi, Amir
PY - 2022/04/01
Y2 - 2023/06/07
TI - Closed ideals in the functionally countable subalgebra of C(X)
JF - Applied General Topology
JA - Appl. Gen. Topol.
VL - 23
IS - 1
SE -
DO - 10.4995/agt.2022.15844
UR - http://ojs.upv.es/index.php/AGT/article/view/15844
SP - 79-90
AB - <p>In this paper, closed ideals in C<sub>c</sub>(X), the functionally countable subalgebra of C(X), with the m<sub>c</sub>-topology, is studied. We show that if<br />X is CUC-space, then C<sup>*</sup><sub>c</sub>(X) with the uniform norm-topology is a Banach algebra. Closed ideals in C<sub>c</sub>(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 C<sub>c</sub>(X) is closed if and only if it is an intersection of maximal ideals of C<sub>c</sub>(X). It is also shown that every ideal in C<sub>c</sub>(X) with the m<sub>c</sub>-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<sup>*</sup><sub>c</sub>(X) is an intersection of maximal ideals of C<sup>*</sup><sub>c</sub>(X) if and only if X is pseudo compact. Finally, we show that if X is a P-space, then the family of e<sub>c</sub>-ultrafilters and z<sub>c</sub>-ultrafilter coincide.</p><p> </p>
ER -