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

Document Type: Article dedicated to Uffe Haagerup

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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  ${\mathrm C}^\ast(\Sigma)$.

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