Advances in Operator Theory

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

Marcel de Jeu and Jun Tomiyama

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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 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 ${\mathrm C}^*(\Sigma)$.

Article information

Adv. Oper. Theory, Volume 3, Number 1 (2018), 42-52.

Received: 14 February 2017
Accepted: 8 March 2017
First available in Project Euclid: 5 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46K99: None of the above, but in this section
Secondary: 46H10: Ideals and subalgebras 47L65: Crossed product algebras (analytic crossed products) 54H20: Topological dynamics [See also 28Dxx, 37Bxx]

involutive Banach algebra enveloping $\mathrm{C}^*$-algebra ideal topological dynamical system


Jeu, Marcel de; Tomiyama, Jun. The closure of ideals of $\ell^1(\Sigma)$ in its enveloping $\mathrm{C}^*$-algebra. Adv. Oper. Theory 3 (2018), no. 1, 42--52. doi:10.22034/aot.1702-1116.

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