Banach Representations and Affine Compactifications of Dynamical Systems [chapter]

Eli Glasner, Michael Megrelishvili
2013 Fields Institute Communications  
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
more » ... 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 [8, 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. [44] 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 semigroups below are assumed to be admissible. Given x ∈ X, its orbit is the set Sx = {sx : s ∈ S} and the closure of this set, cl (Sx), is the orbit closure of x. A point x with cl (Sx) = X is called a transitive point, and the set of transitive points is denoted by X tr . We say that the system is point-transitive when X tr = ∅. The system is called minimal if X tr = X. 1.2. Representations of dynamical systems. A representation of a semigroup S on a normed space V is a co-homomorphism h : S → Θ(V ), where Θ(V ) := {T ∈ L(V ) : ||T || ≤ 1} and h(e) = id V . Here L(V ) is the space of continuous linear operators V → V and id V is the identity operator. This is equivalent to the requirement that h : S → Θ(V ) op be a monoid homomorphism, where Θ(V ) op is the opposite semigroup of Θ(V ). If S = G, is a group then h(G) ⊂ Iso (V ), where Iso (V ) is the group of all linear isometries from V onto V . The adjoint operator adj : L(V ) → L(V * ) induces an injective co-homomorphism adj : Θ(V ) → Θ(V * ), adj(s) = s * . We will identify adj(L(V )) and the opposite semigroup L(V ) op ; as well as adj(Θ(V )) ⊂ L(V * ) and its opposite semigroup Θ(V ) op . Mostly we use the same symbol s instead of s * . Since Θ(V ) op acts from the right on V and from the left on V * we sometimes write vs for h(s)(v) and sψ for h(s) * (ψ). A pair of vectors (v, ψ) ∈ V × V * defines a function (called a matrix coefficient of h) m(v, ψ) : S → R, s → ψ(vs) = vs, ψ = v, sψ . The weak operator topology on Θ(V ) (similarly, on Θ(V ) op ) is the weak topology generated by all matrix coefficients. So h : S → Θ(V ) op is weakly continuous iff m(v, ψ) ∈ C(S) for every (v, ψ) ∈ V × V * . The strong operator topology on Θ(V ) (and on Θ(V ) op ) is the pointwise topology with respect to its left (respectively, right) action on the Banach space V . Lemma 1.4. Let h : S → Θ(V ) be a weakly continuous co-homomorphism.Then for every v ∈ V the following map
doi:10.1007/978-1-4614-6406-8_6 fatcat:ywockiwpbbdbhko2rubqfv5vva