  Our aim is to study weak star continuous representations of semigroup actions
into the duals of ``good'' (e.g., reflexive and Asplund) Banach spaces. This
approach leads to flow analogs of Eberlein and Radon-Nikodym compacta and a new
class of functions (Asplund functions) which intimately is connected with
Asplund representations and includes the class of weakly almost periodic
functions. We show that a flow is weakly almost periodic iff it admits
sufficiently many reflexive representations. One of the main technical tools in
this paper is the concept of fragmentability (which actually comes from Namioka
and Phelps) and widespreadly used in topological aspects of Banach space
theory. We explore fragmentability as ``a generalized equicontinuity'' of
flows. This unified approach allows us to obtain several dynamical
applications. We generalize and strengthen some results of Akin-Auslander-Berg,
Shtern, Veech-Troallic-Auslander and Hansel-Troallic. We establish that
frequently, for linear G-actions, weak and strong topologies coincide on, not
necessarily closed, G-minimal subsets. For instance such actions are
``orbitwise Kadec``.
