Jeu, M., Tomiyama, J. (2018). The closure of ideals of $\ell^1(\Sigma)$ in its enveloping $\mathrm{C}^*$-algebra. Advances in Operator Theory, 3(1), 42-52. doi: 10.22034/aot.1702-1116

Marcel de Jeu; Jun Tomiyama. "The closure of ideals of $\ell^1(\Sigma)$ in its enveloping $\mathrm{C}^*$-algebra". Advances in Operator Theory, 3, 1, 2018, 42-52. doi: 10.22034/aot.1702-1116

Jeu, M., Tomiyama, J. (2018). 'The closure of ideals of $\ell^1(\Sigma)$ in its enveloping $\mathrm{C}^*$-algebra', Advances in Operator Theory, 3(1), pp. 42-52. doi: 10.22034/aot.1702-1116

Jeu, M., Tomiyama, J. The closure of ideals of $\ell^1(\Sigma)$ in its enveloping $\mathrm{C}^*$-algebra. Advances in Operator Theory, 2018; 3(1): 42-52. doi: 10.22034/aot.1702-1116

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

Receive Date: 14 February 2017,
Revise Date: 07 March 2017,
Accept Date: 08 March 2017

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 ${\mathrm C}^\ast(\Sigma)$.