# Bounded cocycles: von Neumann algebras and amenability.

 Title: Bounded cocycles: von Neumann algebras and amenability. Author: Bates, Teresa. Abstract: In a 1993 preprint Guyan Robertson proved that every uniformly bounded representation of a discrete group on a finite von Neumann algebra is similar to a unitary representation. We have since discovered that this result was first proved in a paper of Vasilescu and Zsido, published in 1974 (VZ). In this thesis we generalise this result for discrete groupoids, proving that every uniformly bounded cocycle into a finite von Neumann algebra is cohomologous to a unitary cocycle. The corresponding result for cocycles into finite dimensional algebras was proved in (Zim3). We also derive some equivalent definitions of amenability of group actions and provide a new proof of a result of Zimmer regarding uniformly bounded cocycles on amenable G-spaces. We develop some machinery in order to prove these results. This is the theory of ${\cal G}$-flows in which we explore the actions of groupoids on Borel fields of sets. Our development of this theory follows that of the usual theory of flows from topological dynamics. Date: 1995 URI: http://hdl.handle.net/10393/10278

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