Functional Analysis Master Thesis Seminars (Lise Wouters and Bram Vancraeynest)

13h30 - 14h20 : Lise Wouters

14h30 - 15h20 : Bram Vancraeynest

Titles and abstracts

Lise Wouters : Measure equivalence of locally compact groups

Recently Koivisto, Kyed and Raum extended the notion of measure equivalence (introduced by Gromov for discrete groups) to locally compact groups. Examples include lattices (or more generally, closed subgroups of finite covolume) in the same locally compact group. For discrete groups, the notion of measure equivalence is equivalent to the two groups admitting stably orbit equivalent, essentially free, ergodic, pmp actions. The definition for locally compact groups is equivalent to a natural generalization of this condition. In my thesis I have studied in detail which kind of free, ergodic, pmp actions occur for a given group. More specifically, I have studied which types and flow of weights occur for the von Neumann algebra associated to cross section equivalence relations of these actions. It turns out the answer depends on the modular function of the group. As a corollary, I obtained a new proof of a recent result by Koivisto, Kyed and Raum, that up to measure equivalence there are precisely three classes of amenable groups: the compact, noncompact unimodular, and non-unimodular ones. In the seminar I will introduce measure equivalence of locally compact groups, explain the results I have obtained for actions of these groups and how they lead to the corollary. Depending on the available time, I will also discuss some more examples and properties of measure equivalence.

Bram Vancraeynest : Nuclear dimension

Sometimes one thinks of the theory of C*-algebras as "non-commutative topology". This is justified by the relation between abelian C*-algebras and locally compact Hausdorff spaces. Winter and Zacharias introduced the concept of nuclear dimension that brings the notion of dimension to a C*-algebra setting. In the commutative case, the nuclear dimension of the C*-algebra C(X) agrees with the topological dimension of the locally compact Hausdorff space X. Thus we can view nuclear dimension as a "non-commutative dimension". In addition the introduction of nuclear dimension meant a breakthrough in the classification of C*-algebras. In my thesis I studied several properties of nuclear dimension and in this seminar I will discuss them. In particular I will show that the zero dimensional C*-algebras are exactly the approximately finite dimensional (AF) C*-algebras. I will also look at some examples of C*-algebras and their nuclear dimension.