Mathematics – Operator Algebras
Scientific paper
2009-02-04
J. Funct. Anal. 262 (2012) 4746-4765
Mathematics
Operator Algebras
19 pages. Some typos corrected and minor changes in presentation. Final version, to appear in Journal of Functional Analysis
Scientific paper
Given a topological dynamical system $\Sigma = (X, \sigma)$, where $X$ is a compact Hausdorff space and $\sigma$ a homeomorphism of $X$, we introduce the associated Banach $^*$-algebra crossed product $\ell^1 (\Sigma)$ and analyse its ideal structure. This algebra is the Banach algebra most naturally associated with the dynamical system, and it has a richer structure than its well studied $C^*$-envelope, as becomes evident from the possible existence of non-self-adjoint closed ideals. This paper initiates the study of these algebras and links their ideal structure to the topological dynamics. It is determined when exactly the algebra is simple, or prime, and when there exists a non-self-adjoint closed ideal. In addition, a structure theorem is obtained for the case when $X$ consists of one finite orbit, and the algebra is shown to be Hermitian if $X$ is finite. The key to these results lies in analysing the commutant of $C(X)$ in the algebra, which can be shown to be a maximal abelian subalgebra with non-zero intersection with each non-zero closed ideal.
Jeu Marcel de
Svensson Christian
Tomiyama Jun
No associations
LandOfFree
On the Banach $*$-algebra crossed product associated with a topological dynamical system does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with On the Banach $*$-algebra crossed product associated with a topological dynamical system, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the Banach $*$-algebra crossed product associated with a topological dynamical system will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-553354