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Winter 2018 The closure of ideals of $\ell^1(\Sigma)$ in its enveloping $\mathrm{C}^*$-algebra
Marcel de Jeu, Jun Tomiyama
Adv. Oper. Theory 3(1): 42-52 (Winter 2018). 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 ${\mathrm C}^*(\Sigma)$, the enveloping $\mathrm{C}^*$-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 ${\mathrm C}^*(\Sigma)$ is again a proper two-sided ideal of ${\mathrm C}^*(\Sigma)$.

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

Information

Received: 14 February 2017; Accepted: 8 March 2017; Published: Winter 2018
First available in Project Euclid: 5 December 2017

zbMATH: 1385.46032
MathSciNet: MR3730338
Digital Object Identifier: 10.22034/aot.1702-1116

Subjects:
Primary: 46K99
Secondary: 46H10, 47L65, 54H20‎

Rights: Copyright © 2018 Tusi Mathematical Research Group

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Vol.3 • No. 1 • Winter 2018
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