  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 $L(G) \subseteq M \rtimes_\alpha G$,
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 $L(G)
\subseteq M \rtimes_\alpha G$ 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 Furstenberg-Zimmer distal tower for an ergodic action on a
probability space, and we establish new versions of the Furstenberg-Zimmer
structure theorems 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.
