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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 isnaturally 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-sidedideal of $\ell^1(\Sigma)$ in ${\mathrm C}^\ast(\Sigma)$ is again a propertwo-sided ideal of ${\mathrm C}^\ast(\Sigma)$.

The closure of ideals of $\boldsymbol{\ell^1(\Sigma)}$ in its enveloping $\boldsymbol{\mathrm{C}^\ast}$-algebra