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

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 $\mathrm{C}(X){⋊}_{\sigma }\mathbb{Z}$ 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 ${\mathrm{C}}^{\ast }(\Sigma )$.

9 pages. Minor changes in presentation from the original. Final version; to appear in Adv. Oper. Theory

References in corpus (1)

• Algebraically irreducible representations and structure space of the Banach algebra associated with a topological dynamical system