Banach representations and affine compactifications of dynamical systems

TL;DR

This paper explores the structure of Banach spaces through associated affine semigroups, linking properties like being Asplund or Rosenthal to the topological nature of these semigroups, and studies their role in dynamical systems.

Contribution

It introduces the concept of E(V) for Banach spaces, characterizes spaces via the properties of E(V), and investigates the relationship between enveloping semigroups and affine compactifications in dynamical systems.

Findings

Separable Banach space V is Asplund iff E(V) is metrizable.

V is Rosenthal iff E(V) is a Rosenthal compactum.

Distal non-equicontinuous systems lack E-compatible compactifications.

Abstract

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.