On Real Algebras Associated with Ideal Convergence

Abstract

In this paper we will prove some properties of the algebras $W(I)$, $\;c^*(I)$ and $c(I)$ which arise when considering the notion of $I$-convergence of real sequences. By $I$-convergence of a real sequence we mean the convergence according to the filter associated to an ideal $I$ on the set $\mathbb{N}$. $W(I)$ is the set of sequences having finite $I$-variation, $ c^*$ and $c$ stand for two kinds of $I$-convergence. In particular, we are studying the maximal ideals of $W(I)$ in the cases where $I$ is admissible or non-admissible. We do that by using the topological properties of the semi-normed space $l^{\infty}(I)$. Finally, we discuss the question whether all maximal ideals $P$ of $W(I)$ are of the form $P=c_0(\mathcal{M})\cap W(I)$ where $\mathcal{M}$ is a maximal ideal of $2^{\mathbb{N}}$.