Banach Representations and Affine Compactifications of Dynamical Systems

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.1.1. Semigroups and actions. Let S be a semigroup which is also a topological space. By λ a : S → S, x → ax and ρ a : S → S, x → xa we denote the left and right a-transitions. The subset Λ(S) := {a ∈ S : λ a is continuous} is called the topological center of S. Definition 1.1. A semigroup S as above is said to be:(1) a right topological semigroup if every ρ a is continuous.(Let A be a subsemigroup of a right topological semigroup S. If A ⊂ Λ(S) then the closure cl(A) is a right topological semigroup. In general, cl(A) is not necessarily a subsemigroup of S (even if S is compact right topological and A is a left ideal). Also Λ(S) may be empty for general compact right topological semigroup S. See [7, p. 29].Definition 1.2. Let S be a semitopological semigroup with a neutral element e. Let π : S × X → X be a left action of S on a topological space X. This means that ex = x and s 1 (s 2 x) = (s 1 s 2 )x for all s 1 , s 2 ∈ S and x ∈ X, where as usual, we write sx instead of π(s, x) = λ s (We say that X is a dynamical S-system (or an S-space or an S-flow ) if the action π is separately continuous (that is, if all orbit maps ρ x : S → X and all translations λ s : X → X are continuous). We sometimes write it as a pair (S, X).A right system (X, S) can be defined analogously. If S op is the opposite semigroup of S with the same topology then (X, S) can be treated as a left system (S op , X) (and vice versa). Fact 1.3. [43]Let G be aČech-complete (e.g., locally compact or completely metrizable) semitopological group. Then every separately continuous action of G on a compact space X is continuous.Notation: All semigroups S are assumed to be monoids, i.e, semigroups with a neutral element which will be denoted by e. Also actions are monoidal (meaning ex = x, ∀x ∈ X) and separately continuous. We reserve the symbol G for the case when S is a group. All right topological semig...

A Note on the Topological Group c0

“…This question is closely related to the setting of this work. Indeed, by [4] (Prop. 6.13) (resp., by [4] (Cor.…”

“…Indeed, by [4] (Prop. 6.13) (resp., by [4] (Cor. 6.20)) the metrizability (first countability) of P guarantees that the corresponding algebra is a subset of Asp(G) (resp.…”

“…One of the most useful references about semigroup compactifications is a book of Berglund, Junghenn and Milnes [9]. For some new directions (regarding topological groups) see also [3,4,15,16].…”

“…Hence, P is metrizable. If we assume that γ is a semigroup compactification then the separability of A f guarantees by [4] ( Prop. 6.13) that A f ⊂ Asp(G).…”

“…The WAP triviality of H + [0, 1] was conjectured by Pestov. Recall also that for G := H + [0, 1] every Asplund (hence also every WAP) function is constant and every continuous representation G → Iso(V) on an Asplund (hence also reflexive) space V must be trivial [3]. In contrast one may show (see [4,5]) that H + [0, 1] is representable on a (separable) Rosenthal space (a Banach space is Rosenthal if it does not contain a subspace topologically isomorphic to l 1 ).…”