Tusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X3120180101The closure of ideals of $ell^1(Sigma)$ in its enveloping $mathrm{C}^*$-algebra42524404710.22034/aot.1702-1116ENMarcel deJeuJun TomiyamaJournal Article20170214If $X$ is a compact Hausdorff space and $sigma$ is a homeomorphism of $X$, then an involutive Banach algebra $ell^1(Sigma)$ of crossed product type is naturally associated with the topological dynamical system $Sigma=(X,sigma)$. We initiate the study of the relation between two-sided ideals of $ell^1(Sigma)$ and ${mathrm C}^ast(Sigma)$, the enveloping $mathrm{C}^ast$-algebra ${mathrm C}(X)rtimes_sigma<br />mathbb Z$ of $ell^1(Sigma)$. Among others, we prove that the closure of a proper two-sided ideal of $ell^1(Sigma)$ in ${mathrm C}^ast(Sigma)$ is again a proper two-sided ideal of ${mathrm C}^ast(Sigma)$.http://www.aot-math.org/article_44047_f65a8f1062ea283744db5848a9363ba9.pdf