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Joinings and relative ergodic properties of W*-dynamical systems
Thesis (PhD)--University of Pretoria, 2019.
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University of Pretoria
2020
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| _version_ | 1867613517798965248 |
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| access_status_str | Open Access |
| author2 | Duvenhage, Rocco |
| author_browse | Duvenhage, Rocco |
| author_facet | Duvenhage, Rocco |
| collection | Thesis |
| dc_rights_str_mv | © 2019 University of Pretoria. All rights reserved. The copyright in this work vests in the University of Pretoria. No part of this work may be reproduced or transmitted in any form or by any means, without the prior written permission of the University of Pretoria. |
| description | Thesis (PhD)--University of Pretoria, 2019. |
| format | Thesis |
| id | oai:repository.up.ac.za:2263/73237 |
| institution | University of Pretoria (South Africa) |
| language | English |
| last_indexed | 2026-06-10T12:37:24.815Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UPSpace — University of Pretoria Institutional Repository |
| publishDate | 2020 |
| publishDateRange | 2020 |
| publishDateSort | 2020 |
| publisher | University of Pretoria |
| publisherStr | University of Pretoria |
| record_format | dspace |
| source_str | UPSpace — University of Pretoria Institutional Repository |
| spelling | oai:repository.up.ac.za:2263/73237 Joinings and relative ergodic properties of W*-dynamical systems Duvenhage, Rocco malcolmbruceking@gmail.com Ströh, Anton King, Malcolm Bruce UCTD Noncommutive ergodic theory Thesis (PhD)--University of Pretoria, 2019. We prove a characterization of relative weak mixing in W*-dynamical systems in terms of a relatively independent joining. We then define a noncommutative version of relative discrete spectrum, show that it generalizes both the classical and noncommutative absolute cases and give examples. Chapter 1 reviews the GNS construction for normal states, the related semicyclic representation on von Neumann algebras, Tomita-Takasaki theory and conditional expectations. This will allow us to define, in the tracial case, the basic construction of Vaughan Jones and its associated lifted trace. Dynamics is introduced in the form of automorphisms on von Neumann algebras, represented using the cyclic and separating vector and then extended to the basic construction. In Chapter 2, after introducing a relative product system, we discuss relative weak mixing in the tracial case. We give an example of a relative weak mixing W*-dynamical system that is neither ergodic nor asymptotically abelian, before proving the aforementioned characterization. Chapter 3 defines relative discrete spectrum as complementary to relative weak mixing. We motivate the definition using work from Chapter 2. We show that our definition generalizes the classical and absolute noncommutative case of isometric extensions and discrete spectrum, respectively. The first example is a skew product of a classical system with a noncommutative one. The second is a purely noncommutative example of a tensor product of a W*-dynamical system with a finite-dimensional one. Pilot Programme Top-Up Bursary, Department of Mathematics and Applied Mathematics, University of Pretoria. Mathematics and Applied Mathematics PhD Unrestricted 2020-02-12T09:36:04Z 2020-02-12T09:36:04Z 2020-04-15 2019 Thesis King, MB 2019, Joinings and relative ergodic properties of W*-dynamical systems, PhD Thesis, University of Pretoria, Pretoria, viewed yymmdd <http://hdl.handle.net/2263/73237> A2020 http://hdl.handle.net/2263/73237 en © 2019 University of Pretoria. All rights reserved. The copyright in this work vests in the University of Pretoria. No part of this work may be reproduced or transmitted in any form or by any means, without the prior written permission of the University of Pretoria. application/pdf University of Pretoria |
| spellingShingle | UCTD Noncommutive ergodic theory Joinings and relative ergodic properties of W*-dynamical systems |
| title | Joinings and relative ergodic properties of W*-dynamical systems |
| title_full | Joinings and relative ergodic properties of W*-dynamical systems |
| title_fullStr | Joinings and relative ergodic properties of W*-dynamical systems |
| title_full_unstemmed | Joinings and relative ergodic properties of W*-dynamical systems |
| title_short | Joinings and relative ergodic properties of W*-dynamical systems |
| title_sort | joinings and relative ergodic properties of w dynamical systems |
| topic | UCTD Noncommutive ergodic theory |
| url | http://hdl.handle.net/2263/73237 |