On the commutant of C(X) in C*-crossed products by Z and their representations

Research output: Contribution to journalArticle

Abstract

For the C*-crossed product C*(Sigma) associated with an arbitrary topological dynamical system Sigma = (X, sigma), we provide a detailed analysis of the commutant, in C*(Sigma), of C(X) and the commutant of the image of C(X) under an arbitrary Hilbert space representation (pi) over tilde of C*(E), In particular, we give a concrete description of these commutants, and also determine their spectra. We show that, regardless of the system E, the commutant of C(X) has non-zero intersection with every non-zero, not necessarily closed or self-adjoint, ideal of C*(Z). We also show that the corresponding statement holds true for the commutant of (pi) over tilde (C(X)) tinder the assumption that a certain family of pure states of (pi) over tilde (C*(Z)) is total. Furthermore we establish that, if C(X) subset of C(X)', there exist both a C*-Kibalgebra properly between C(X) and C(X)' which has the aforementioned intersection property, and such a C*-subalgebra which does not have this properly. We also discuss existence of* a projection of norm one from C*(Sigma) onto the commutant of C(X). (c) 2009 Elsevier Inc. All rights reserved.

Details

Authors
  • Christian Svensson
  • Jun Tomiyama
Organisations
Research areas and keywords

Subject classification (UKÄ) – MANDATORY

  • Mathematics

Keywords

  • Commutant, Ideals, Crossed product, Dynamical system, subalgebra, Maximal abelian
Original languageEnglish
Pages (from-to)2367-2386
JournalJournal of Functional Analysis
Volume256
Issue number7
Publication statusPublished - 2009
Publication categoryResearch
Peer-reviewedYes