On the Banach *-algebra crossed product associated with a topological dynamical system

Research output: Contribution to journalArticle

Abstract

Given a topological dynamical system Sigma = (X, sigma), where X is a compact Hausdorff space and a a homeomorphism of X, we introduce the Banach *-algebra crossed product l(1) (E) most naturally associated with Sigma and initiate its study. It has a richer structure than its well investigated C*-envelope, as becomes evident from the possible existence of non-self-adjoint closed ideals. We link its ideal structure to the dynamics, determining when the algebra is simple, or prime, and when there exists a non-self-adjoint closed ideal. A structure theorem is obtained when X consists of one finite orbit, and the algebra is shown to be Hermitian if X is finite. The key lies in analysing the commutant of C(X) in the algebra, which is shown to be a maximal abelian subalgebra with non-zero intersection with each non-zero closed ideal. (C) 2012 Elsevier Inc. All rights reserved.

Details

Authors
  • Marcel de Jeu
  • Christian Svensson
  • Jun Tomiyama
Organisations
Research areas and keywords

Subject classification (UKÄ) – MANDATORY

  • Mathematics

Keywords

  • Involutive Banach algebra, Crossed product, Ideal structure, Topological, dynamical system
Original languageEnglish
Pages (from-to)4746-4765
JournalJournal of Functional Analysis
Volume262
Issue number11
Publication statusPublished - 2012
Publication categoryResearch
Peer-reviewedYes