Banach Representations and Affine Compactifications of Dynamical Systems

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 nonsensitive) 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.