Fragmentability and Continuity of Semigroup Actions

Trans. Amer. Math. Soc.

2012

107

Section: Fact 14 ([16])mentioning

confidence: 99%

“…Radon-Nikodým (RN) if it admits a faithful representation on an Asplund Banach space [30,13]. If G = {1}, we get the class of Radon-Nikodým compact spaces in the sense of Namioka [33].…”

Section: Asplund Representations Rn and Hns Systems A Dynamical Sysmentioning

confidence: 99%

See 1 more Smart Citation

Representations of dynamical systems on Banach spaces not containing $l_{1}$

Trans. Amer. Math. Soc.

2012

107

“…The main question considered in [26] was whether the dual action π * of G on V * is jointly continuous with respect to the norm topology on V * . When this is the case we say that the action π (and, also the corresponding representation h : G → Iso(V ), when π is an action by linear isometries) is adjoint continuous.…”

Section: Representations Of Groups and G-spaces On Banach Spacesmentioning

confidence: 99%

“…In general, not every continuous representation is adjoint continuous (see for example [26]). A standard example is the representation of the circle group G := T on V := C(T) by translations.…”

Section: Representations Of Groups and G-spaces On Banach Spacesmentioning

confidence: 99%

New algebras of functions on topological groups arising from G-spaces

2008

Fund. Math.

Abstract. For a topological group G we introduce the algebra SU C(G) of strongly uniformly continuous functions. We show that SU C(G) contains the algebra W AP (G) of weakly almost periodic functions as well as the algebras LE(G) and Asp(G) of locally equicontinuous and Asplund functions respectively. For the Polish groups of order preserving homeomorphisms of the unit interval and of isometries of the Urysohn space of diameter 1, we show that SU C(G) is trivial. We introduce the notion of fixed point on a class P of flows (P -f pp) and study in particular groups with the SU C-f pp. We study the Roelcke algebra (= U C(G) = right and left uniformly continuous functions) and SUC compactifications of the groups S(N), of permutations of a countable set, and H(C), the group of homeomorphisms of the Cantor set. For the first group we show that W AP (G) = SU C(G) = U C(G) and also provide a concrete description of the corresponding metrizable (in fact Cantor) semitopological semigroup compactification. For the second group, in contrast, we show that SU C(G) is properly contained in U C(G). We then deduce that for this group U C(G) does not yield a right topological semigroup compactification.

Banach Representations and Affine Compactifications of Dynamical Systems

Asymptotic Geometric Analysis

2013

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...