Banach Journal of Mathematical Analysis

Noncommutative spectral synthesis for the involutive Banach algebra associated with a topological dynamical system

Marcel de Jeu and Jun Tomiyama

Full-text: Open access


If $\Sigma=(X,\sigma)$ is a topological dynamical system, where $X$ is a compact Hausdorff space and $\sigma$ is a homeomorphism of $X$, then a crossed product involutive Banach algebra $\ell^1$ is naturally associated with these data. If $X$ consists of one point, then $\ell^1$ is the group algebra of the integers, hence the general$\ell^1$could be regarded as a noncommutative $\ell^1$-algebra. In this paper, we study spectral synthesis for the closed ideals of $\ell^1$ in two versions, one modeled after $C(X)$and one modeled after $\ell^1(\mathbb{Z})$. We identify the closed ideals which are equal to (what is the analogue of) the kernel of their hull, and determine when this holds for all closed ideals, i.e., when spectral synthesis holds. In both models, this is the case precisely when $\Sigma$ is free.

Article information

Banach J. Math. Anal., Volume 7, Number 2 (2013), 103 -135 .

First available in Project Euclid: 20 March 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46K99
Secondary: 46H10 47L65

Involutive Banach algebra crossed product structure of ideals spectral synthesis topological dynamical system


de Jeu , Marcel; Tomiyama , Jun. Noncommutative spectral synthesis for the involutive Banach algebra associated with a topological dynamical system. Banach J. Math. Anal. 7 (2013), no. 2, 103 --135. doi:10.15352/bjma/1363784226.

Export citation


  • F.F. Bonsall and J. Duncan, Complete normed algebras, Springer, 1973.
  • J. Dixmier, $C^\ast$-algebras, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.
  • E.G. Effros and F. Hahn, Locally compact transformation groups and $C^\ast$-algebras, Mem. Amer. Math. Soc. no. 75, 1967.
  • E.C. Gootman and J. Rosenberg, The structure of crossed product $C^\ast$-algebras: a proof of the generalized Effros-Hahn conjecture, Invent. Math. 52 (1979), 283–298.
  • M. de Jeu, C. Svensson and J. Tomiyama, On the Banach \supast-algebra crossed product associated with a topological dynamical system, J. Funct. Anal. 262 (2012), 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), 47–75.
  • E. Kaniuth, A course in commutative Banach algebras, Springer, New York, 2009.
  • R. Larsen, Banach algebras. An introduction, Marcel Dekker, New York, 1973.
  • P. Malliavin, Impossibilité de la synthèse spectrale sur les groups abéliens non compacts, Publ. Math. Inst. Hautes Etudes Sci. Paris, 1959, 61–68.
  • W.R. Rudin, Fourier analysis on groups, Interscience Publishers, New York-London, 1962.
  • C. Svensson and J. Tomiyama, On the commutant of $\coeffalg$ in $C^\ast$-crossed products by $Z$ and their representations, J. Funct. Anal. 256 (2009), 2367–2386.
  • J. Tomiyama, Invitation to $C^*$-algebras and topological dynamics, World Scientific Advanced Series in Dynamical Systems, Vol. 3, World Scientific Publishing Co., Singapore, 1987.
  • J. Tomiyama, The interplay between topological dynamics and theory of $C^*$-algebras, Lecture Notes Series, Vol. 2, Seoul National University Research Institute of Mathematics Global Analysis Research Center, Seoul, 1992.
  • J. Tomiyama, The interplay between topological dynamics and theory of $C^\ast$-algebras. II, Research publication notes from the Research Inst. of Math. Sci. of Kyoto University, Vol. 1151, Kyoto University, Kyoto, 2000.
  • J. Tomiyama, Structure of ideals and isomorphisms of $C^\ast$-crossed products by single homeomorphisms, Tokyo J. Math. 23 (2000), 1–13.
  • J. Tomiyama, Hulls and kernels from topological dynamical systems and their applications to homeomorphism $C^\ast$-algebras, J. Math. Soc. Japan 56 (2004), no. 2, 349–364.
  • J. Tomiyama, Classification of ideals of homeomorphism $C^\ast$-algebras and quasidiagonality of their quotient algebras, Acta Appl. Math. 108 (2009), 561–572.