Title: The closure of ideals of $\boldsymbol{\ell^1(Σ)}$ in its enveloping $\boldsymbol{\mathrm{C}^\ast}$-algebra,
Abstract: 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 ${\mathrm C}(X)\rtimes_\sigma \mathbb Z$ 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 ${\mathrm C}^\ast(\Sigma)$.