Quasinormalizers in crossed products of von Neumann algebras

We study the relationship between the dynamics of the action alpha of a discrete group G on a von Neumann algebra M, and structural properties of the associated crossed product inclusion L(G) subset of M kappa alpha G, and its intermediate subalgebras. This continues a thread of research originating in classical structural results for ergodic actions of discrete, abelian groups on probability spaces. A key tool in the setting of a noncommutative dynamical system is the set of quasinormalizers for an inclusion of von Neumann algebras. We show that the von Neumann algebra generated by the quasinormalizers captures analytical properties of the inclusion L(G) subset of M kappa alpha G such as the Haagerup Approximation Property, and is essential to capturing "almost periodic" behavior in the underlying dynamical system. Our von Neumann algebraic point of view yields a new description of the Furstenb erg-Zimmer distal tower for an ergodic action on a probability space, and we establish new versions of the Furstenb erg-Zimmer structure theorem for general, tracial W*-dynamical systems. We present a number of examples contrasting the noncommutative and classical settings which also build on previous work concerning singular inclusions of finite von Neumann algebras. (c) 2024 Elsevier Inc. All rights reserved.