  To every Banach space V we associate a compact right topological affine
semigroup E(V). We show that a separable Banach space V is Asplund if and only
if E(V) is metrizable, and it is Rosenthal (i.e. it does not contain an
isomorphic copy of $l_1$) if and only if E(V) is a Rosenthal compactum. We
study representations of compact right topological semigroups in E(V). In
particular, representations of tame and HNS-semigroups arise naturally as
enveloping semigroups of tame and HNS (hereditarily non-sensitive) dynamical
systems, respectively. As an application we obtain a generalization of a
theorem of R. Ellis. A main theme of our investigation is the relationship
between the enveloping semigroup of a dynamical system X and the enveloping
semigroup of its various affine compactifications Q(X). When the two coincide
we say that the affine compactification Q(X) is E-compatible. This is a
refinement of the notion of injectivity. We show that distal non-equicontinuous
systems do not admit any E-compatible compactification. We present several new
examples of non-injective dynamical systems and examine the relationship
between injectivity and E-compatibility.
