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
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.