%0 Journal Article
%T The closure of ideals of $ell^1(Sigma)$ in its enveloping $mathrm{C}^*$-algebra
%J Advances in Operator Theory
%I Tusi Mathematical Research Group (TMRG)
%Z 2538-225X
%A Jeu, Marcel de
%A Tomiyama, Jun
%D 2018
%\ 01/01/2018
%V 3
%N 1
%P 42-52
%! The closure of ideals of $ell^1(Sigma)$ in its enveloping $mathrm{C}^*$-algebra
%K Primary 46K99
%K Secondary 46H10, 47L65, 54H20
%R 10.22034/aot.1702-1116
%X If $X$ 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 ${mathrm C}(X)rtimes_sigmamathbb Z$ 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 ${mathrm C}^ast(Sigma)$.
%U http://www.aot-math.org/article_44047_f65a8f1062ea283744db5848a9363ba9.pdf