Ec-Filters and ec-ideals in the functionally countable subalgebra of C*(X)

Abstract

<p>The purpose of this article is to study and investigate e<sub>c</sub>-filters on X and e<sub>c</sub>-ideals in C<sup>*</sup><sub>c </sub>(X) in which they are in fact the counterparts of z<sub>c</sub>-filters on X and z<sub>c</sub>-ideals in C<sub>c</sub>(X) respectively. We show that the maximal ideals of C<sup>*</sup><sub>c </sub>(X) are in one-to-one correspondence with the e<sub>c</sub>-ultrafilters on X. In addition, the sets of e<sub>c</sub>-ultrafilters and z<sub>c</sub>-ultrafilters are in one-to-one correspondence. It is also shown that the sets of maximal ideals of C<sub>c</sub>(X) and C<sup>*</sup><sub>c </sub>(X) have the same cardinality. As another application of the new concepts, we characterized maximal ideals of C<sup>*</sup><sub>c </sub>(X). Finally, we show that whether the space X is compact, a proper ideal I of C<sub>c</sub>(X) is an e<sub>c</sub>-ideal if and only if it is a closed ideal in C<sub>c</sub>(X) if and only if it is an intersection of maximal ideals of C<sub>c</sub>(X).</p>