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Crossed Product Structures Associated with Topological Dynamical Systems

Svensson, Christian LU (2009) In Doctoral Theses in Mathematical Sciences 2009:1.
Abstract
We study connections between topological dynamical systems and associated algebras of crossed product type. We derive equivalences between structural properties of a crossed product and dynamical properties of the associated system and furthermore derive qualitative results concerning the crossed product that are true regardless of the corresponding dynamical system. The systems principally investigated are pairs of a compact Hausdorff space and a homeomorphism, where the integers act on former via iterations of the latter.

With such a system a crossed product C*-algebra can be associated. We do not only focus on the C*-crossed product of a system, but also on a Banach *-algebra and a non-complete *-algebra that can both be... (More)
We study connections between topological dynamical systems and associated algebras of crossed product type. We derive equivalences between structural properties of a crossed product and dynamical properties of the associated system and furthermore derive qualitative results concerning the crossed product that are true regardless of the corresponding dynamical system. The systems principally investigated are pairs of a compact Hausdorff space and a homeomorphism, where the integers act on former via iterations of the latter.

With such a system a crossed product C*-algebra can be associated. We do not only focus on the C*-crossed product of a system, but also on a Banach *-algebra and a non-complete *-algebra that can both be embedded by *-isomorphisms as dense subalgebras of the C*-algebra; the C*-crossed product is the so-called enveloping

C*-algebra of this Banach *-algebra. While investigations of the connections between a system and its C*-algebra have an extensive history, considerations of the other two algebras are new. For these algebras, we derive analogues of results from the case of

C*-algebras, but also prove a theorem whose counterpart in the

C*-algebra case is false. Furthermore we study the interplay between crossed products of Banach algebras by the integers and naturally associated systems. (Less)
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author
supervisor
opponent
  • Professor Eilers, Soren, Department of Mathematical Sciences, University of Copenhagen, Denmark
organization
publishing date
type
Thesis
publication status
published
subject
keywords
maximal abelian subalgebra, commutant, C*-algebra, dynamical system, ideal, Banach algebra, Crossed product
in
Doctoral Theses in Mathematical Sciences
volume
2009:1
pages
109 pages
publisher
Centre for Mathematical Sciences, Lund University
defense location
Lecture Hall MH:C, Centre for Mathematical Sciences, Sölvegatan 18, Lund university, Faculty of Engineering
defense date
2009-04-01 13:15:00
ISSN
1404-0034
ISBN
978-91-628-7709-5
project
Non-commutative Analysis of Dynamics, Fractals and Wavelets
language
English
LU publication?
yes
id
85cd5ba1-66b8-4c4e-8c18-c69ca6a4a0fe (old id 1304182)
date added to LUP
2016-04-01 13:36:51
date last changed
2019-05-21 13:39:28
@phdthesis{85cd5ba1-66b8-4c4e-8c18-c69ca6a4a0fe,
  abstract     = {{We study connections between topological dynamical systems and associated algebras of crossed product type. We derive equivalences between structural properties of a crossed product and dynamical properties of the associated system and furthermore derive qualitative results concerning the crossed product that are true regardless of the corresponding dynamical system. The systems principally investigated are pairs of a compact Hausdorff space and a homeomorphism, where the integers act on former via iterations of the latter.<br/><br>
With such a system a crossed product C*-algebra can be associated. We do not only focus on the C*-crossed product of a system, but also on a Banach *-algebra and a non-complete *-algebra that can both be embedded by *-isomorphisms as dense subalgebras of the C*-algebra; the C*-crossed product is the so-called enveloping <br/><br>
C*-algebra of this Banach *-algebra. While investigations of the connections between a system and its C*-algebra have an extensive history, considerations of the other two algebras are new. For these algebras, we derive analogues of results from the case of <br/><br>
C*-algebras, but also prove a theorem whose counterpart in the <br/><br>
C*-algebra case is false. Furthermore we study the interplay between crossed products of Banach algebras by the integers and naturally associated systems.}},
  author       = {{Svensson, Christian}},
  isbn         = {{978-91-628-7709-5}},
  issn         = {{1404-0034}},
  keywords     = {{maximal abelian subalgebra; commutant; C*-algebra; dynamical system; ideal; Banach algebra; Crossed product}},
  language     = {{eng}},
  publisher    = {{Centre for Mathematical Sciences, Lund University}},
  school       = {{Lund University}},
  series       = {{Doctoral Theses in Mathematical Sciences}},
  title        = {{Crossed Product Structures Associated with Topological Dynamical Systems}},
  volume       = {{2009:1}},
  year         = {{2009}},
}