The closure of ideals of in its enveloping -algebra

If $X$ is a compact Hausdorff space and $\sigma$ is a homeomorphism of $X$, then an involutive Banach algebra ${\mathrm{\ell }}^1(\mathrm{\Sigma })$ of crossed product type is naturally associated with the topological dynamical system $\mathrm{\Sigma }=(X,\sigma )$. We initiate the study of the relation between two-sided ideals of ${\mathrm{\ell }}^1(\mathrm{\Sigma })$ and ${\mathrm{C}}^{\ast }(\mathrm{\Sigma })$, the enveloping ${\mathrm{C}}^{\ast }$-algebra $\mathrm{C}(X){⋊}_{\sigma }\mathbb{Z}$ of ${\mathrm{\ell }}^1(\mathrm{\Sigma })$. Among others, we prove that the closure of a proper two-sided ideal of ${\mathrm{\ell }}^1(\mathrm{\Sigma })$ in ${\mathrm{C}}^{\ast }(\mathrm{\Sigma })$ is again a proper two-sided ideal of ${\mathrm{C}}^{\ast }(\mathrm{\Sigma })$.