Representation of spectrums of $C^{\ast}$-algebras of bounded functions in terms of filters

Abstract

Let $X$ be a non-empty set and let $\mathcal{F}$ be any $C^{\ast}$-subalgebra of $\ell ^{\infty}(X)$ containing the constant functions. We show that the spectrum of $\mathcal{F}$ can be considered as a space of certain filters determined by $\mathcal{F}$ on $X$. Furthermore, we show that these filters describe the topology of the spectrum of $\mathcal{F}$.