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

In this dissertation we develop aspects of ergodic theory for C*-dynamical systems for which the C*-algebras are allowed to be noncommutative. We define four ergodic properties, with analogues in classic ergodic theory, and study C*-dynamical systems possessing these properties. Our analysis will show that, as in the classical case, only certain combinations of these properties are permissable on C*-dynamical systems. In the second half of this work, we construct concrete noncommutative C*-dynamical systems having various permissable combinations of the ergodic properties. This shows that, as in classical ergodic theory, these ergodic properties continue to be meaningful in the noncommutative case, and can be useful to classify and analyse C*-dynamical systems. / Dissertation (MSc)--University of Pretoria, 2013. / gm2014 / Mathematics and Applied Mathematics / unrestricted

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