Unique ergodicity in C*-dynamical systems

Abstract

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.