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

## TL;DR

This paper explores the relationship between ideals of a Banach algebra associated with a dynamical system and its enveloping C*-algebra, proving that proper ideals remain proper upon closure.

## Contribution

It establishes that the closure of proper two-sided ideals of the Banach algebra in its C*-algebra is also proper, advancing understanding of their structural relationship.

## Key findings

- Closure of proper ideals remains proper in the C*-algebra
- Relationship between ideals of Banach algebra and C*-algebra clarified
- Initiates study of ideal correspondence in crossed product algebras

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