The aim of this dissertation is to investigate ergodic properties, in particular
unique ergodicity, in a noncommutative setting, that is in C*-dynamical
systems. Fairly recently Abadie and Dykema introduced a broader notion
of unique ergodicity, namely relative unique ergodicity. Our main focus
shall be to present their result for arbitrary abelian groups containing a
F lner sequence, and thus generalizing the Z-action dealt with by Abadie
and Dykema, and also to present examples of C*-dynamical systems that
exhibit variations of these (uniquely) ergodic notions.
Abadie and Dykema gives some characterizations of relative unique ergodicity,
and among them they state that a C*-dynamical system that is
relatively uniquely ergodic has a conditional expectation onto the xed point
space under the automorphism in question, which is given by the limit of
some ergodic averages. This is possible due to a result by Tomiyama which
states that any norm one projection of a C*-algebra onto a C*-subalgebra
is a conditional expectation. Hence the rst chapter is devoted to the proof
of Tomiyama's result, after which some examples of C*-dynamical systems
are considered.
In the last chapter we deal with unique and relative unique ergodicity
in C*-dynamical systems, and look at examples that illustrate these notions.
Speci cally, we present two examples of C*-dynamical systems that
are uniquely ergodic, one with an R2-action and the other with a Z-action,
an example of a C*-dynamical system that is relatively uniquely ergodic but
not uniquely ergodic, and lastly an example of a C*-dynamical system that
is ergodic, but not uniquely ergodic. / Dissertation (MSc)--University of Pretoria, 2013. / gm2014 / Mathematics and Applied Mathematics / unrestricted
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:up/oai:repository.up.ac.za:2263/40335 |
| Date | January 2013 |
| Creators | Van Wyk, Daniel Willem |
| Contributors | Duvenhage, Rocco, danie.vanwyk@up.ac.za, Stroh, Anton |
| Publisher | University of Pretoria |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Dissertation |
| Rights | © 2013 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. |
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