# The Halmos-von Neumann theorem

** Speaker:** Patrick Hermle, University of Wuppertal

** Date:** April 13, 2023

** Abstract:** A classical problem of ergodic theory is to determine whether given measure preserving systems are isomorphic. The Halmos-von Neumann theorem solves this problem for ergodic systems with discrete spectrum, i. e. the unimodular eigenspaces of the Koopman operator are total in L2. It states that two such systems are isomorphic if and only if the point spectrum of the induced Koopman operators coincide. We present a topological proof of this theorem using topological models. Then we generalize this theorem to G-systems by replacing the acting group Z by an arbitrary topological group G and using tools of abstract harmonic analysis (e. g. representatation theory, Tannaka-Krein duality). This is a joint work with Henrik Kreidler (University of Wuppertal).