Thesis (Ph.D.(Mathematics and Applied Mathematics))--University of Pretoria, 2008. / Includes bibliographical references.

We prove a characterization of relative weak mixing in W*-dynamical systems in terms of a relatively independent joining. We then define a noncommutative version of relative discrete spectrum, show that it generalizes both the classical and noncommutative absolute cases and give examples. Chapter 1 reviews the GNS construction for normal states, the related semicyclic representation on von Neumann algebras, Tomita-Takasaki theory and conditional expectations. This will allow us to define, in the tracial case, the basic construction of Vaughan Jones and its associated lifted trace. Dynamics is introduced in the form of automorphisms on von Neumann algebras, represented using the cyclic and separating vector and then extended to the basic construction. In Chapter 2, after introducing a relative product system, we discuss relative weak mixing in the tracial case. We give an example of a relative weak mixing W*-dynamical system that is neither ergodic nor asymptotically abelian, before proving the aforementioned characterization. Chapter 3 defines relative discrete spectrum as complementary to relative weak mixing. We motivate the definition using work from Chapter 2. We show that our definition generalizes the classical and absolute noncommutative case of isometric extensions and discrete spectrum, respectively. The first example is a skew product of a classical system with a noncommutative one. The second is a purely noncommutative example of a tensor product of a W*-dynamical system with a finite-dimensional one. / Thesis (PhD)--University of Pretoria, 2019. / Pilot Programme Top-Up Bursary, Department of Mathematics and Applied Mathematics, University of Pretoria. / Mathematics and Applied Mathematics / PhD / Unrestricted

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This thesis studies orbit equivalence in the almost continuous setting. Recently A. del Junco and A. Sahin obtained an almost continuous version of Dye’s theorem. They proved that any two ergodic measure-preserving homeomorphisms of Polish spaces are almost continuously orbit equivalent. One purpose of this thesis is to extend their result to all free actions of countable amenable groups. We also show that the cocycles associated with the constructed orbit equivalence are almost continuous. In the second part of the thesis we obtain an analogue of Dye’s reconstruction theorem for etale equivalence relations in the almost continuous setting. We introduce topological full groups of etale equivalence relations and show that if the topological full groups are isomorphic, then the equivalence relations are almost continuously orbit equivalent.

This thesis studies orbit equivalence in the almost continuous setting. Recently A. del Junco and A. Sahin obtained an almost continuous version of Dye’s theorem. They proved that any two ergodic measure-preserving homeomorphisms of Polish spaces are almost continuously orbit equivalent. One purpose of this thesis is to extend their result to all free actions of countable amenable groups. We also show that the cocycles associated with the constructed orbit equivalence are almost continuous. In the second part of the thesis we obtain an analogue of Dye’s reconstruction theorem for etale equivalence relations in the almost continuous setting. We introduce topological full groups of etale equivalence relations and show that if the topological full groups are isomorphic, then the equivalence relations are almost continuously orbit equivalent.

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