@article {
author = {Jeu, Marcel and Tomiyama, Jun},
title = {The closure of ideals of $\ell^1(\Sigma)$ in its enveloping $\mathrm{C}^*$-algebra},
journal = {Advances in Operator Theory},
volume = {3},
number = {1},
pages = {42-52},
year = {2018},
publisher = {Tusi Mathematical Research Group (TMRG)},
issn = {2538-225X},
eissn = {2538-225X},
doi = {10.22034/aot.1702-1116},
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 ${\mathrm C}^\ast(\Sigma)$.},
keywords = {Primary 46K99,Secondary 46H10, 47L65, 54H20},
url = {http://www.aot-math.org/article_44047.html},
eprint = {http://www.aot-math.org/article_44047_f65a8f1062ea283744db5848a9363ba9.pdf}
}