## Advances in Operator Theory

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.aot/1512497951

Digital Object Identifier
doi:10.22034/aot.1702-1116

Mathematical Reviews number (MathSciNet)
MR3730338

Zentralblatt MATH identifier
1385.46032

#### Citation

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. https://projecteuclid.org/euclid.aot/1512497951

#### References

• F. F. Bonsall and J. Duncan, Complete normed algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 80, Springer-Verlag, New York-Heidelberg, 1973.
• M. de Jeu, C. Svensson, and J. Tomiyama, On the Banach *-algebra crossed product associated with a topological dynamical system, J. Funct. Anal. 262 (2012), no. 11, 4746–4765.
• M. de Jeu and J. Tomiyama, Maximal abelian subalgebras and projections in two Banach algebras associated with a topological dynamical system, Studia Math. 208 (2012), no. 1, 47–75.
• M. de Jeu and J. Tomiyama, Noncommutative spectral synthesis for the involutive Banach algebra associated with a topological dynamical system, Banach J. Math. Anal. 7 (2013), no. 2, 103–135.
• M. de Jeu and J. Tomiyama, Algebraically irreducible representations and structure space of the Banach algebra associated with a topological dynamical system, Adv. Math. 301 (2016), 79–115.
• J. Dixmier, $\mathrm{C}^*$-algebras, North-Holland Mathematical Library, vol. 15, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Translated from the French by Francis Jellett.
• N. Jacobson, A topology for the set of primitive ideals in an arbitrary ring, Proc. Nat. Acad. Sci. U. S. A. 31 (1945), 333–338.
• J. Tomiyama, The interplay between topological dynamics and theory of $\mathrm{C}^*$-algebras, Lecture Notes Series, vol. 2, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1992.
• D. P. Williams, Crossed products of $\mathrm{C}^*$-algebras, Mathematical Surveys and Monographs, vol. 134, American Mathematical Society, Providence, RI, 2007.