Descriptive Properties of Measure Preserving Actions and the Associated Unitary Representations

Abstract

This thesis consists of two independent parts. In the first part, we study the descriptive complexity of full groups [E]. The main result is i) If E is not smooth, then [E] is Sigma⁰₃3 complete; ii) If E is smooth, then [E] is closed. In the second part, we study descriptive properties of the Koopman unitary repreesentation associated with the measure preserving action. We characterize the smoothness and compressibility of the equivalence induced by the unitary representaion. We also study many connections between the equivalence relation on L^2(X) and the equivalence relation on X.