M.Sc Thesis
M.Sc Student Lessel Alon Compact Models for Group Actions Department of Mathematics Professor Uri Bader

Abstract

Given a measurable action of a locally compact, second-countable group on a Lebesgue space, it is easy to define an action of the same group on the space's W*-algebra (or Von-Neumann algebra) of essentially bounded measurable functions. Conversely, given a group action on an abstract separable and commutative W*-algebra, one may ask whether or not it can be realized as an action on a Lebesgue space. This is known to be true, and in fact, it is known that this Lebesgue space can be chosen to be a compact metric space, called a compact model for the group action. In this work, we extend this to countable families of separable commutative W*-algebras (or equivalently, Lebesgue spaces), showing that compact models exist for each space such that the group action on the compact models is continuous and that the induced maps between the models become continuous maps. For this we use tools from operator algebras. The work also includes a discussion of the equivalence between this approach and the approach based on Boolean algebras.