Representations of dynamical systems on Banach spaces not containing $l_1$

For a topological group G, we show that a compact metric G-space is tame if and only if it can be linearly represented on a separable Banach space which does not contain an isomorphic copy of $l_1$ (we call such Banach spaces, Rosenthal spaces). With this goal in mind we study tame dynamical systems and their representations on Banach spaces.

1. Introduction 1.1. Some important dichotomies. Rosenthal's celebrated dichotomy theorem asserts that every bounded sequence in a Banach space either has a weak Cauchy subsequence or a subsequence equivalent to the unit vector basis of l 1 (an l 1 -sequence). Consequently a Banach space V does not contain an l 1 -sequence if and only if every bounded sequence in V has a weak-Cauchy subsequence [40]. In the present work we will call a Banach space satisfying these equivalent conditions a Rosenthal space.
The theory of Rosenthal spaces is one of the cases where the interplay between Analysis and Topology gives rise to many deep results. Our aim is to show the relevance of Topological Dynamics in this interplay. In particular, we examine representability of dynamical systems on Rosenthal spaces and show that being tame is a complete characterization of these systems.
First we recall several results and ideas. The following dichotomy is a version of a result of Bourgain, Fremlin and Talagrand [3] (as presented in the book of Todorcević [45], Proposition 1 of Section 13). Fact 1.1 (BFT dichotomy). Let X be a Polish space and let {f n } ∞ n=1 ⊂ C(X) be a sequence of real valued functions which is pointwise bounded (i.e. for each x ∈ X the sequence {f n (x)} ∞ n=1 is bounded in R). Let K be the pointwise closure of {f n } ∞ n=1 in R X . Then either K ⊂ B 1 (X), where B 1 (X) denotes the space of all real valued Baire 1 functions on X (i.e. K is a Rosenthal compact) or K contains a homeomorphic copy of βN.
In [15,Theorem 3.2] the following dynamical dichotomy, in the spirit of Bourgain-Fremlin-Talagrand theorem, was established. Fact 1.2 (A dynamical BFT dichotomy). Let (G, X) be a metric dynamical system and let E(X) be its enveloping semigroup. We have the following dichotomy. Either (1) E(X) is separable Rosenthal compact, hence with cardinality card E(X) ≤ 2 ℵ 0 ; or (2) the compact space E contains a homeomorphic copy of βN, hence card E(X) = 2 2 ℵ 0 .
In [12] a metric dynamical system is called tame if the first alternative occurs, i.e. E(X) is a Rosenthal compact. By [3] every Rosenthal compact is a Fréchet space (and in particular its topology is determined by the converging sequences). Thus, either E(X) (although not necessarily metrizable) has a nice topological structure, or it is as unruly as possible containing a copy of βN. As to the metrizability of E(X), recent results [15] and [17] assert that E(X) is metrizable if and only if the metric compact G-space X is hereditarily non-sensitive (HNS), if and only if X is Asplund representable (see Subsection 6.2).

1.2.
Main results and related facts. One of the main results of the present work is the following: Theorem 1. 3. Let X be a compact metric G-space. The following conditions are equivalent: (1) (G, X) is a tame G-system.
This theorem continues a series of recent results which link dynamical properties of G-systems (like WAP and HNS) to their representability on "good" Banach spaces (Reflexive and Asplund respectively). See Sections 1.3 and 6 below for more details.
One of the important questions in Banach space theory until the mid 70's was to construct a separable Rosenthal space which is not Asplund. The first counterexamples were constructed independently by James [23] and Lindenstrauss [29]. In view of Theorem 1. 3 we now see that a fruitful way of producing such distinguishing examples comes from dynamical systems. Just consider a compact metric tame G-system which is not HNS (see e.g. Remark 7.11 below) and then apply Theorem 1. 3.
In order to get a better perspective on the position of tame systems in the hierarchy of dynamical systems we remind the reader of some enveloping semigroup characterizations. For a recent review of enveloping semigroup theory we refer to [14]. It is well known that a compact G-space X is WAP (weakly almost periodic) if and only if its enveloping semigroup E(X) ⊂ X X consists of continuous maps [9]. Recently the following characterization of tameness was established.

Fact 1.4. [17] A compact metric dynamical system (G, X) is tame if and only if every element of E(X) is a Baire 1 function (equivalently, has a point of continuity property) from X to itself.
A function f : X → Y has a point of continuity property if for every closed nonempty subset A of X the restriction f | A : A → Y has a point of continuity. For compact X and metrizable Y it is equivalent to the fragmentability (see Section 2 and Lemma 2.5) of the function f . The topological concept of fragmentability comes in fact from Banach space theory (Jayne-Rogers). For dynamical applications of fragmentability we refer to [30,31,32,15]. Fact 1.4 suggests the following general definition: Definition 1.5. Let X be a (not necessarily metrizable) compact G-space. We say that X is tame if for every element p ∈ E(X) the function p : X → X is fragmented.
The class of tame dynamical systems contains the class of HNS systems and hence also WAP systems. Indeed, as we already mentioned, every function p : X → X (p ∈ E(X)) is continuous for WAP systems. As to the HNS systems they can be characterized as those G-systems where the family of maps {p : X → X} p∈E(X) is a fragmented family (see Fact 6.5 and Definition 2.7 below). In particular, every individual p : X → X is a fragmented map. Thus, these enveloping semigroup characterizations yield a natural hierarchy of the three classes, WAP, HNS and Tame, dynamical systems.
In [28] Köhler introduced the definition of regularity for cascades (i.e. Zdynamical systems) in terms of independent sequences and, using results of Bourgain-Fremlin-Talagrand has shown that her definition can be reformulated in terms of l 1 -sequences. Extending Köhler's definition to arbitrary topological groups G we say that compact G-space X is regular if, for any f ∈ C(X), the orbit f G does not contain an l 1 -sequence (in other words the second alternative is ruled out in the Rosenthal dichotomy). As we will see later, in Corollary 7.9, a G-system is regular if and only if it is tame (for metrizable X this fact was mentioned in [12]).
In Theorem 8.10 we give a characterization of Rosenthal representable G-systems. As a particular case (for trivial G) we get a topological characterization of compact spaces which are homeomorphic to weak * compact subsets in the dual of Rosenthal spaces. A well known result characterizes Rosenthal spaces as those Banach spaces whose dual has the weak Radon-Nikodým property [44,. It is therefore natural to call such a compact space a weakly Radon-Nikodým compactum (WRN). Theorem 8.5 gives a remarkably simple characterization in terms of fragmentability. Namely, a compact space X is WRN if and only if there exists a subset F ⊂ C(X) such that F separates points of X and the pointwise closure of F in R X consists of fragmented maps from X to R.
Theorem 8.10 is strongly related to yet another important characterization of Rosenthal Banach spaces. Precisely, let V be a Banach space with dual V * and second dual V * * . One may look at elements of V * * as functions on the unit weak star compact ball B * := B V * ⊂ V * . While the elements of V are clearly continuous on B * it is not true in general for elements from V * * . By a result of Odell and Rosenthal [37], a separable Banach space V is Rosenthal iff every element v * * from V * * is a Baire one function on B * . More generally E. Saab and P. Saab [43] show that V is Rosenthal iff every element of V * * has a point of continuity property when restricted to B * . Equivalently, every restriction of v * * to a bounded subset M is fragmented as a function (M, w * ) → R (see Fact 3.11 below). This result emphasizes once more the relevance of fragmentability concepts in this theory (see also Proposition 3.12).
Answering a question of Talagrand [44,, R. Pol [38] gave an example of a separable compact Rosenthal space K which cannot be embedded in B 1 (X) for any compact metrizable X. We say that a compact space K is strongly Rosenthal if it is homeomorphic to a subspace of B 1 (X) for a compact metrizable X. We say that a compact space K is an admissible Rosenthal compactum if there exists a metrizable compact space X and a subset Z ⊂ C(X) such that the pointwise closure cls p (Z) of Z in R X consists of Baire 1 functions and K ⊂ cls p (Z). Clearly every admissible compactum is strongly Rosenthal. We do not know whether these two classes of compact spaces coincide. Note that the enveloping semigroup K := E(X) of a compact metrizable G-space X is admissible iff (G, X) is tame (Proposition 9.4). The following related, purely topological, result is another consequence of our analysis. Theorem 1.6. Let K be a compact space. The following conditions are equivalent: (1) K is a weak-star closed bounded subset in the second dual of a separable Rosenthal Banach space V . (2) K is an admissible Rosenthal compactum. Remark 1.7. We note that the main results of our work remain true for semigroup actions once some easy modifications are introduced.
Remark 1.8. The attentive reader will not fail to detect the major importance to our work of the papers [5], [3], and the book [44].
1.3. The hierarchy of Banach representations. In the following table we encapsulate some features of the trinity: dynamical systems, enveloping semigroups, and Banach representations. Let X be a compact metrizable G-space and E(X) denote the corresponding enveloping semigroup. The symbol f stands for an arbitrary function in C(X) and f G = {f • g : g ∈ G} denotes its orbit.

Dynamical characterization
Enveloping semigroup Banach representation WAP cls (f G) is a subset of C(X) Every element is continuous Reflexive Every element is Baire 1 Rosenthal Table 1. The hierarchy of Banach representations

Fragmented maps and families
The following definition is a generalized version of fragmentability in the sense of Jayne and Rogers [25]. Implicitly it already appears in a paper of Namioka and Phelps [36]. For the case of functions see also [24].
Definition 2.1. [30] Let (X, τ ) be a topological space and (Y, µ) a uniform space. We say that X is (τ, µ)-fragmented by a (not necessarily continuous) function f : X → Y if for every nonempty subset A of X and every ε ∈ µ there exists an open subset O of X such that O ∩ A is nonempty and the set f (O ∩ A) is ε-small in Y . We also say in that case that the function f is fragmented. Notation: f ∈ F(X, Y ), whenever the uniformity µ is understood If Y = R then we write simply F(X).

Remarks 2.2.
(1) In Definition 2.1.1 when Y = X, f = id X and µ is a metric uniform structure, we get the usual definition of fragmentability [25].
(2) It is enough to check the condition of Definition 2.1 only for closed subsets A ⊂ X and for ε ∈ µ from a subbase γ of µ (that is, the finite intersections of the elements of γ form a base of the uniform structure µ). (3) Namioka's joint continuity theorem [34] implies that every weakly compact subset K of a Banach space is (weak,norm)-fragmented (that is, id K : (K, weak) → (K, norm) is fragmented). (4) Recall that a Banach space V is an Asplund space if the dual of every separable Banach subspace is separable, iff every bounded subset A of the dual V * is (weak * ,norm)-fragmented, iff V * has the Radon-Nikodým property. Reflexive spaces and spaces of the type c 0 (Γ) are Asplund. For more details cf. [4,11,35].

(5) A Banach space V is Rosenthal if and only if every bounded subset
A of the dual V * is (weak * topology, weak uniformity)-fragmented. This follows by Proposition 3.12.
Recall that f : X → Y is barely continuous, [33], if for every nonempty closed subset A ⊂ X, the restricted map f ↾ A has at least one point of continuity. Following [44,Section 14] the set of barely continuous functions f : X → R is denoted by B ′ r (X). Lemma 2.3.
(1) Every barely continuous f is fragmented.
(3): Consider the weak uniformity µ w on Y generated by the system {f i : Y → Z i } i∈I . Since this system separates the points and each Z i is a Hausdorff uniform space we get that µ w is a Hausdorff uniformity on Y . Furthermore µ w is continuous on Y . Now it is clear that µ w coincides with the unique compatible uniformity on the compact space Y . The system of entourages  [35]. (5): If f 0 is fragmented then f is fragmented by (2). If f is fragmented then f 0 is fragmented by (4) (with Y 1 = Y 2 ). (6): For a fixed ε > 0 consider The fragmentability implies that O ε is dense in X. Clearly {O 1 n : n ∈ N} serves as the required dense G δ subset of X.
2.1. Baire class one functions. Given two topological spaces X and Y , a function f : X → Y is of Baire class 1 or more briefly Baire 1 if the inverse image of every open set in Y is F σ (the union of countably many closed sets) in X. In general a Baire 1 function need not be the same as a limit of a sequence of continuous functions. The following results are well known. Mainly they are classical and come from R. Baire. See for example [6,26,17].

Lemma 2.4.
(1) If Y is metrizable and {f n : X → Y } n∈N is a sequence of continuous functions converging pointwise to f : X → Y then f is Baire 1.
(2) If X is separable and metrizable then a real valued function f : X → R is Baire 1 iff f is a pointwise limit of a sequence of continuous functions.
We denote by B 1 (X) the space of all real valued Baire 1 functions on X equipped with the pointwise convergence topology. That is, B 1 (X) is a topological subspace of the product space R X .
As usual, a space is Baire if the intersection of any countable family of dense open sets is dense. hereditarily Baire means that every closed subspace is a Baire space. Lemma 2.5. Let (X, τ ) be a hereditarily Baire (e.g., Polish, or compact) space, (Y, ρ) a pseudometric space. Consider the following assertions: Proof. For (a) ⇔ (b) use Lemma 2.3. The equivalence (b) ⇔ (c) for Polish X and separable Y is well known (see [26,Theorem 24.15]) and goes back to Baire.

2.2.
Fragmented families. The following definition was introduced in [15] and independently in the Ph.D. Thesis of M.M. Guillermo [21] (we thank Cascales for pointing out this reference).
(1) We say that a family of functions F = {f : (X, τ ) → (Y, µ)} is fragmented if the condition of Definition 2.1.1 holds simultaneously for all f ∈ F . That is, f (O ∩ A) is ε-small for every f ∈ F . It is equivalent to say that the mapping is (τ, µ U )-fragmented, where µ U is the uniform structure of uniform convergence on the set Y F of all mappings from F into (Y, µ).
(2) Analogously one can define the notion of a barely continuous family.
The latter means that every closed nonempty subset A ⊂ X contains a point a ∈ A such that If µ is pseudometrizable then so is µ U . Therefore if in addition (X, τ ) is hereditarily Baire then it follows by Proposition 2.5.1 that F is a fragmented family if and only if F is a barely continuous family.
Fragmented families, like equicontinuous families, are stable under pointwise closures as the following lemma shows.
(1) Let F be compact, X isČech-complete, M metrizable and X × F → M separately continuous. Then F is a barely continuous (and hence fragmented) family of maps on X.
(2) Let F be compact metrizable, X Polish, M separable metrizable. assume that X × F → M is a map such that every x : F → M continuous, y : X → R is continuous at every y ∈ Y dense subset in F . Then F is a barely continuous (and hence fragmented) family.
(2): Since every x : F → M continuous, the natural map j : X → C(cls p (F ), M ) is well defined. The function x → j(x)(f ) = f (x) is continuous for every f ∈ F . For every closed subset X 0 ⊂ X the induced map j| X 0 : X 0 → C(cls p (F ), M ) has a point of continuity property by virtue of [17,Proposition 2.3]. Therefore, cls p (F ) is a fragmented family by Definition 2.7.2. Hence also its subfamily F is a fragmented family. Definition 2.10. We say that a family of functions F = {f : (X, τ ) → (Y, µ)} is sub-fragmented if every sequence in F has a subsequence which is a fragmented family on X.
Example 2.11. Let V be a Banach space. Then we can treat B V as a family of functions on the weak * compact space B V * .
(1) B V is a fragmented family of functions on B V * if and only if V is Asplund. This fact easily follows from the following well known characterization of Asplund spaces: V is Asplund iff B V * is (weak * ,norm)fragmented (Remark 2.2.4).
(2) B V is a sub-fragmented family of functions on B V * if and only if V is a Rosenthal Banach space (see Proposition 3.12).

Rosenthal families in C(X) and Rosenthal Banach spaces
Let X be a topological space and A ⊂ X. We say that A is relatively compact in X if the closure cls (A) is a compact subset of X. We say that A is sequentially precompact in X if every infinite sequence in A has a subsequence which converges in X.

Bourgain-Fremlin-Talagrand theorems.
Recall that a topological space K is Rosenthal compact [20] if it is homeomorphic to a pointwise compact subset of the space B 1 (X) of functions of the first Baire class on a Polish space X. All metric compact spaces are Rosenthal. An example of a separable non-metrizable Rosenthal compact is the Helly compact of all (not only strictly) increasing selfmaps of [0, 1] in the pointwise topology. Another is the "two arrows" space of Alexandroff and Urysohn. Recall that a topological space K is Fréchet if for every A ⊂ K and every x ∈ cls (A) there exists a sequence of elements of A which converges to x.
The following theorem is due to Bourgain-Fremlin-Talagrand [3, Theorem 3F], generalizing a result of Rosenthal. The second assertion (BFT dichotomy) is presented as in the book of Todorcević [45] (see Proposition 1 of Section 13). Rosenthal compact) or K contains a homeomorphic copy of βN.
Clearly, βN the Stone-Čech compactification of the natural numbers N, is not Fréchet, and hence it cannot be embedded into a Rosenthal compact space.
The following important result is a part of [44,Theorem 14.1.7] in Talagrand's book. Here in order to reformulate assertion (3) recall that, by Corollary 2.6, F(X) = B 1 (X) = B ′ r (X) holds for every Polish space X. (1) F does not contain a subsequence equivalent to the unit basis of l 1 .
(2) Each sequence of F has a pointwise convergent subsequence in R X (i.e., F is sequentially precompact in R X ).
(4) Every sequence in F has a Cauchy subsequence with respect to the weak uniformity µ w generated by the collection of all maps {x : Proof. The equivalence of (1), (2) and (3)  (4) ⇔ (2): Let f n be an infinite sequence in F . Then its subsequence f n k is a µ w -Cauchy subsequence iff this subsequence is pointwise convergent in R X .
Let F ⊂ C(X) be a norm bounded subset. Then the pointwise closure cls p (F ) in R X is compact. The following lemma examines four natural conditions expressing "smallness" of a family F ⊂ C(X).
Proof. (a) ⇒ (b): Let Y be a dense countable subset of X. Since every function φ ∈ cls p (F ) is continuous we get that the natural continuous projection R X → R Y induces an injection on cls p (F ). Since cls p (F ) is compact we get its homeomorphic embedding into the second countable space The family F is fragmented means by Definition 2.7 that the natural map X → R F is fragmented, where R F carries the uniformity of uniform convergence. Then the image of X is separable as it follows by [15,Lemma 6.5]. Now [35,Theorem 4.1] implies that the pointwise closure Since F is a fragmented family its pointwise closure cls p (F ) is again a fragmented family (see Lemma 2.8). In particular, every member φ ∈ cls p (F ) is a fragmented map on X. Since X is Polish this means that φ ∈ B 1 (X).
Later in Proposition 7.10 we consider the particular case of an orbit F := f G, where f is a continuous function on a compact metrizable G-space X.
Fact 3.2 suggests the following definition.
Definition 3.4. Let X be a topological space. We say that a subset F ⊂ C(X) is a Rosenthal family (for X) if F is norm bounded and Proposition 3.5. Let X be a compact space and F a bounded subset of C(X). The following conditions are equivalent: F is a Rosenthal family for X).
(2) Every sequence in F has a subsequence which is a fragmented family on X (i.e. F is a sub-fragmented family of maps on X).
If X is metrizable then each of these conditions is equivalent to the following.
Proof. We can reduce the proof to the case of metrizable X. For every sequence S in F we can pass to the associated compact metrizable factor X S of X (as in the proof of Proposition 4.2). Denote by α : X → X S the quotient map. By Lemma 2.3.5, S is (eventually) fragmented family for X iff S is is (eventually) fragmented family for X S . So with no restriction of generality we will assume that X is metrizable.
(1) ⇒ (2): Let S be a sequence in F . By the implication (3) ⇒ (2) from Fact 3.2 we can choose a pointwise convergent subsequence {f n } n∈N with the limit f = lim f n . Denote by K the compact metrizable subset {f n } n∈N ∪ {f } in R X . Consider the evaluation map X × K → R. Then we can apply Lemma 2.9.2 which implies that K (and hence also its subfamily {f n } n∈N ) is a fragmented family of maps on X.
(2) ⇒ (1): Let S be a sequence in F . By our assumption there exists a subsequence {f n } n∈N which is a fragmented family of functions on X. By Lemma 3.3 the pointwise closure cls p ({f n } n∈N ) is a (compact) metrizable subspace in R X . Therefore we can choose a convergent subsequence of Lemma 3.6. Let q : X 1 → X 2 be a map between two topological spaces. Then (1) The natural map phism between the compact spaces cls p (F 2 ) and cls p (F 1 ).
(3)(a): By the continuity of γ we get γ( . Now we can finish the proof of (3)(a) by applying Lemma 2.3.5.

3.2.
Banach spaces not containing l 1 . Definition 3.7. Let us say that a Banach space V is Rosenthal if it does not contain an isomorphic copy of l 1 .
Clearly the class of Asplund spaces is a subclass of the class of Rosenthal spaces. The difference between these two classes can be illustrated in terms of fragmentability. Compare last two items of Remarks 2.2 and Proposition 3.12.
Recall the following famous result of Rosenthal.
Fact 3.8. (Rosenthal [40]) Let V be a Banach space. The following conditions are equivalent: (2) Every bounded sequence in V has a weak-Cauchy subsequence.
Taking into account Definition 3.4 we get the following reformulation of Fact 3.8. Lemma 3.9. Let V be a Banach space. The following conditions are equivalent: Next let us recall some additional characterizations of Rosenthal spaces. First, for the separable case, we have the following theorem. ( Thus a separable Banach space V does not contain an isomorphic copy of l 1 if and only if every element x * * ∈ V * * is Baire 1 when restricted to the unit ball B V * with its weak-star topology σ(V * , V ). This classical result of Odell and Rosenthal was generalized in [43].
Let A be a weak * compact subset of a dual Banach space V * . Following [39] we say that A has the scalar point of continuity property if for each weak * compact subset M of A and every x * * ∈ V * * , the restriction x * * | M of x * * to M has a point of continuity.  (1) V is a Rosenthal Banach space.
(2) Each x * * ∈ V * * is a fragmented map when restricted to the weak * compact ball B V * .
τ w * is the weak * topology and µ w is the weak uniformity on A. (4) B V is a sub-fragmented family of functions on (B V * , w * ).
(1) The equivalence of (1) and (2) in Proposition 3.12 is indeed a natural generalization of the Odell-Rosenthal result because for compact metrizable X we have B 1 (X) = F(X) (Corollary 2.6.2) and the weak * compact ball B V * is metrizable for separable V .
(2) Let V be a Banach space and A a weak * compact absolutely convex subset of V * . Then by [39,Theorem 9], A has the scalar point of continuity property if and only if A is a weak Radon-Nikodym subset (WRN for short). We refer to [4,43,39] for exact definitions and additional information about WRN subsets. See also Theorem 8.5 below.

Further properties of Rosenthal families
4.1. Convex hull. The following important result is proved in Bourgain-Fremplin-Talagrand paper [3].
This result can be generalized for not necessarily metrizable compact spaces replacing B 1 (X) by F(X). At the same time we formulate in terms of Rosenthal families. Proof. First case: If X is a metrizable compact space then combine Fact 4.1 and Proposition 3.5.
Second case: For a general compact space X. Since by Fact 3.2 we have only to examine sequences we can reduce the verification to the case of metrizable compacta X. More precisely, let {y n } n∈N be a sequence in co(F ). There exists a sequence A := {f k } k∈N in F such that y n ∈ co({f k } k∈N ) for every n ∈ N. Denote by X A the canonical quotient of X induced by the collection A. (The equivalence relation on X is defined as Let q A : X → X A be the corresponding quotient map. By construction every y ∈ co(A) preserves the relation ∼ A . Hence y induces a continuous map y A : By our assumption A := {f k } k∈N is a Rosenthal family for X. Then {(f k ) A } k∈N is a Rosenthal family for X A (Use Lemma 3.6.3). Since X A is metrizable we can apply the first case and deduce that the convex hull It follows that by Lemma 3.6.3(b) the collection co({(f k )} k∈N ) (and hence its subcollection {y n } n∈N ) is a Rosenthal family for X.

4.2.
The natural affine extension map T : bB 1 (X) → bB 1 (B * ). For every compact metric space X denote by bB 1 (X) the collection of bounded Baire 1 real valued functions on X. That is, Then bB 1 (X) is a topological subspace of B 1 (X) with respect to the pointwise topology (inherited from R X ). One can define a natural injective map T : bB 1 (X) → bB 1 (B * ), where B * , as before, is the weak * compact unit ball of C(X) * . We will use Riesz representation theorem and Lebesgue's Dominated Convergence Theorem.
Each f ∈ bB 1 (X) is universally measurable for every compact metric space X (see for example [3,Proposition 1F]). That is, for every measure µ ∈ B * we can define This map is well defined. Indeed, first note that when f ∈ C(X) then is the canonical isometric inclusion of the corresponding Banach spaces and ·, · : C(X) × C(X) * → R is the canonical bilinear mapping. Now if f ∈ bB 1 (X) then f is a pointwise limit of a sequence of continuous functions h n ∈ C(X) (Lemma 2.4. (2)). Since f : X → R is a bounded function we can assume in addition that the sequence h n is uniformly bounded. By Lebesgue's Convergence Theorem it follows that T (f ) is a pointwise limit of the sequence T (h n ) = i(h n ), n ∈ N. Since every i(h n ) ∈ C(B * ) we conclude by Lemma 2.4. (2) that Each T (f ) for f ∈ bB 1 (X) can be treated as an element of the second dual C(X) * * of C(X). Moreover the pointwise topology of B 1 (B * ) and the weak * -topology on C(X) * * agree on T (B 1 (X)). Proof. Lebesgue's Convergence Theorem implies that T is sequentially continuous. The boundedness of T (A) is easy.
Proposition 4.5. If F ⊂ C(X) is a Rosenthal family for a compact metric space X then the restriction of T on cls p (F ) induces a homeomorphism Proof. As F is a Rosenthal family for X its pointwise closure cls p (F ) is a compact subset of B 1 (X). Moreover, cls p (F ) is a uniformly bounded subset of bB 1 (X) because F is bounded (Definition 3.4). In view of Lemma 4.4 the restricted map T : cls p (F ) → bB 1 (B * ) is sequentially continuous. By the Bourgain-Fremlin-Talagrand theorem, Fact 3.1.(1), we know that cls p (F ), being a Rosenthal compactum, is Fréchet. For a Fréchet space a sequentially continuous map is continuous and we conclude that the map T : cls p (F ) → bB 1 (B * ) is a continuous injection, and therefore a homeomorphism, of cls p (F ) onto its image in bB 1 (B * ).
Proposition 4.6. Let X be a compact space and F ⊂ C(X). The following conditions are equivalent: (1) F is a Rosenthal family for X.
(2) F is a Rosenthal family for B * .
Proof. We can use Fact 3.2 which depends on sequences only. Let f n be a sequence in F . Since F is a Rosenthal family for X, by Fact 3.2 the sequence f n contains a pointwise convergent subsequence f n k . This subsequence is even weakly convergent (weakly in the Banach space C(X)). Indeed, by Lebesgue's Dominated Convergence Theorem bounded sequences in C(X) are pointwise convergent if and only if they are weakly convergent. Thus by Fact 3.2 we obtain that F is a Rosenthal family for B * .

G-flows and functions
The following definition is standard (for more details see for example [15]).
(1) A function f ∈ C(X) on a G-space X comes from a compact G-system Y if there exist a G-compactification ν : X → Y (so, ν is onto if X is compact) and a function F ∈ C(Y ) such that Then necessarily, f is right uniformly continuous in the sense of [15] (notation: f ∈ RUC(X)) that is, the orbit map f : G → C(X), g → f g is norm continuous.
(2) A function f ∈ RUC(G) comes from a pointed system (Y, y 0 ) if for some continuous function F ∈ C(Y ) we have f (g) = F (gy 0 ), ∀g ∈ G.
Defining ν : X = G → Y by ν(g) = gy 0 observe that this is indeed a particular case of 5.1.1.

5.1.
Cyclic G-flows. As a first motivation note a simple fact about Definition 5.1. For every G-space X a function f : X → R lies in RUC(X) iff it comes from a compact G-flow Y . We can choose Y via the maximal G-compactification G → β G G = Y of G generated by the algebra RUC(X). This is the largest possibility in this setting. Among all possible G-compactifications ν : X → Y of a G-space X such that f comes from (ν, Y ) there exists the smallest one. Namely the smallest closed unital G-subalgebra A f of RUC(X) generated by the orbit f G of f in RUC(X) (note that C(X) = RUC(X) for compact G-spaces X). Denote by X f the Gelfand space |A f | = X A f of the algebra A f . Then the corresponding G-compactification X → Y := X f is the desired one. We call A f and X f the cyclic G-algebra and cyclic G-system of f , respectively. Next we provide an alternative construction and some basic properties of X f . Let X be a (not necessarily compact) G-space. Given f ∈ RUC(X) let I = [− f , f ] ⊂ R and Ω = I G , the product space equipped with the compact product topology. We let G act on Ω by gω(h) = ω(hg), g, h ∈ G.
Define the continuous map and the closure X f := cls (f ♯ (X)) in Ω. Note that X f = f ♯ (X) whenever X is compact.
Denoting the unique continuous extension of f to β G X byf (it exists because f ∈ RU C(X)) we now define a map Let pr e : Ω → R denote the projection of Ω = I G onto the e-coordinate and let F e := pr e ↾ X f : X f → R be its restriction to X f . Thus, F e (ω) := ω(e) for every ω ∈ X f . As before denote by A f the smallest (closed and unital, of course) Ginvariant subalgebra of RUC(X) which contains f . There is then a naturally defined G-action on the Gelfand space X A f = |A f | and a G-compactification (morphism of dynamical systems if X is compact) π f : X → |A f |. Next consider the map π : β G X → |A f |, the canonical extension of π f induced by the inclusion A f ⊂ RUC(X).
The action of G on Ω is not in general continuous. However, the restricted action on X f is continuous for every f ∈ RUC(X). This follows from the second assertion of the next proposition.
(3) f = F e • f ♯ . Thus every f ∈ RUC(X) comes from the system X f . Moreover, if f comes from a system Y and a G-compactification ν : X → Y then there exists a homomorphism α : Y → X f such that f ♯ = α • ν. (4) The orbit F e G = {F e g} g∈G of F e separates points of X f .
Note that if X is compact then in Proposition 5.2, β G X can be replaced simply by X. If X := G with the usual left action then X f is the pointwise closure of the G-orbit Gf := {gf } g∈G of f in RUC(G). Hence (X f , f ) is a transitive pointed G-system.
The cyclic G-systems X f provide "building blocks" for compact G-spaces. That is, every compact G-space can be embedded into the G-product of Gspaces X f . When G is uniformly Lindelöf (e.g. when G is second countable) the compactum X f is metrizable for every compact G-space X and f ∈ C(X). A topological group G is uniformly Lindelöf if for every nonempty open subset O ⊂ G countably many translates g n O cover G. It is well known that G is uniformly Lindelöf iff G is a topological subgroup in a product of second countable groups. If G is uniformly Lindelöf then every compact G-space is embedded into a G-product of metrizable compact G-spaces [15].

Enveloping semigroups.
The enveloping (or Ellis) semigroup E = E(G, X) = E(X) of a dynamical system (G, X) is defined as the closure in X X (with its compact pointwise convergence topology) of the setG = {g : X → X} g∈G of translations considered as a subset of X X . With the operation of composition of maps this is a right topological semigroup. Moreover, the map j = j X : G → E(X), g →g is a right topological semigroup compactification of G. The compact space E(X) becomes a G-space with respect to the natural action Let E = E(X) be the enveloping semigroup of a compact G-system X. For every f ∈ C(X) define Then E f is a pointwise compact subset of R X , being a continuous image of E under the map q f : x ∈ X, f ∈ C(X)}.
(2) Let q : X → X 0 be a continuous onto G-map between compact Gspaces.
(a) There exists a (unique) continuous onto homomorphism Q : Proof. The proof of (1) is straightforward.

Banach representations of flows
Let V be a Banach space. Denote by Iso (V ) the topological group of all linear isometries of V onto itself, equipped with pointwise convergence topology.
A representation (h, α) is faithful will mean that α is a topological embedding.
Every compact G-space admits a canonical faithful representation on the Banach space C(X). A natural question is to characterize dynamical systems according to their representability on nice Banach spaces. 6.1. Reflexive representations and WAP flows. A compact dynamical system (G, X) is weakly almost periodic (WAP) if C(X) = W AP (X). As usual a continuous function f : X → R is WAP if the weakly closure of the orbit f G is weakly compact in C(X). [9]) A compact G-space X is WAP iff every element p ∈ E(X) is a continuous selfmap of X. (1) the dynamical system (G, X) is weakly almost periodic (WAP);

Fact 6.2. (See Ellis and Nerurkar
(2) the dynamical system (G, X) is reflexively representable (that is, admits a faithful representation on a reflexive Banach space). we get the class of Radon-Nikodým compact spaces in the sense of Namioka [35].) One of the main results of [15] was a characterization of RN-systems as those which are "hereditarily non-sensitive" (HNS). To give a precise statement we recall a concept of non-sensitivity (see for instance [18,2,32,15] and the references therein). Let d be a compatible metric on a compact G-system X. We say that (G, X) is non-sensitive if for every ε > 0 there exists a non-empty open set O ⊂ X such that for every g ∈ G the set gO has d-diameter < ε. (This property does not depend on the choice of a compatible metric d.) A system (G, X) is hereditarily non-sensitive (HNS) if all closed G-subsystems are non-sensitive.

Asplund representations, RN and HNS flows. A dynamical system is Radon-Nikodým (RN) if it admits a proper representation on an
For a nonmetrizable version of HNS in terms of uniform structures and some other properties we refer to [15]. Fact 6.4. (See [15] and [17]) Let X be a compact metric G-space. The following conditions are equivalent: (1) the dynamical system (G, X) is hereditarily non-sensitive (HNS); (2) the dynamical system (G, X) is RN, that is, admits a proper representation on an Asplund Banach space; (3) the enveloping semigroup E(X) is metrizable. (1) the dynamical system (G, X) is hereditarily non-sensitive (HNS); (2) the dynamical system (G, X) is RN-approximable, that is, admits sufficiently many representations on Asplund Banach spaces;

Tame dynamical systems and functions
In [15, Theorem 3.2] the following dynamical Bourgain-Fremlin-Talagrand dichotomy was established.  A dynamical BFT dichotomy). Let (G, X) be a metric dynamical system and let E(X) be its enveloping semigroup. We have the following dichotomy. Either (1) E(X) is separable Rosenthal compact, hence with cardinality card E(X) ≤ 2 ℵ 0 ; or (2) the compact space E contains a homeomorphic copy of βN, hence card E(X) = 2 2 ℵ 0 .
In [12] a metric dynamical system is called tame if the first alternative occurs, i.e. E(X) is Rosenthal compact. See also Proposition 9.4 below which shows that E(X) is an admissible Rosenthal compact (in the sense of Definition 9.1).

Fact 7.2. [17] A compact metric dynamical system (G, X) is tame if and only if every element of E(X) is a Baire 1 function (equivalently, fragmented) from X to itself.
This result suggests the following general definition: Definition 7.3. Let X be a (not necessarily metrizable) compact G-space. We say that X is tame if for every element p ∈ E(X) the function p : X → X is fragmented. That is if E(X) ⊂ F(X, X).
We will see later that in fact this class is the same as the class of all regular systems in the sense of Köhler [28]. In particular this gives an enveloping semigroup characterization of regular systems.

Lemma 7.4. Every compact HNS G-space is tame.
Proof. E(X) = {p : X → X} p∈E(X) is a fragmented family when X is HNS by Fact 6.5. In particular we get E(X) ⊂ F(X).
Roughly speaking the difference between HNS and tame systems is just the difference between fragmented families and families which consist of fragmented maps (see Facts 6.4 and 7.2). Proof. The case of subsystems is trivial because the fragmentability of maps is a hereditary property. The cases of products and factors both can be proved using Lemma 2.3.
For factors: let α : X → Y be a G-factor. Lemma 5.3.2 implies that for every p ∈ E(Y ) there exists p X ∈ E(X) such that the following diagram commutes.
Then p X ∈ F(X) because (G, X) is tame. By Lemma 2.3.4 we obtain that p Y ∈ F(Y ). This shows that (G, X) is tame.
For products: let X := i X i be a G-product of compact tame G-spaces X i with canonical G-projections α i : X → X i . For every p ∈ E(X) and every index i we have the following commutative diagram . Then the same is true for α i • p. The family of projections α i separates points of X. Now directly from Lemma 2.3.3 we conclude that p ∈ F(X).
For every G-space X there exists a maximal tame G-compactification (Gfactor if X is compact). If X is a tame G-system then E(X) is also tame as a G-system. Definition 7.6. We say that a continuous function f : X → R on a compact G-space X is tame (notation f ∈ T ame(X)) if it comes from a tame Gsystem. Proposition 7.7. Let X be a compact G-space, f ∈ C(X) and E f = cls p (f G) is the pointwise closure of the orbit of f in R G . The following conditions are equivalent: Proof. The implication (2) ⇒ (1) is obvious by Proposition 5.2 because f comes from the cyclic G-space X f .
(1) ⇒ (3): There exist: a tame compact G-system X 0 , a G-quotient map q : X → X 0 and a function f 0 ∈ C(X 0 ) such that f = f 0 • q. By Lemma 3.6.3 it suffices to show that f 0 G is a Rosenthal family for X 0 . Clearly f 0 G is norm bounded in C(X 0 ). We have to show that the corresponding pointwise closure of . Our G-system X 0 is tame means that every p : X 0 → X 0 (p ∈ E) is fragmented. Thus every f 0 • p is also fragmented (because f 0 is uniformly continuous). So, f 0 G is a Rosenthal family for X 0 .
(3) ⇒ (2): If f G is a Rosenthal family for X then the pointwise closure of the orbit E f := cls p (f G) is a subset of F(X). This means that f • p ∈ F(X) for every p ∈ E. Consider the cyclic G-system X f and the natural Gquotient f ♯ : X → X f . By Proposition 5.2 there exists a continuous function Then also F • gp = F g • p ∈ F(X f ) for every p ∈ E(X f ) and g ∈ G. Now since F G separates points of X f (see Proposition 5.2.4), by Lemma 2.3.3 we conclude that p : X f → X f is a fragmented map for every p ∈ E(X f ). This means that (G, X f ) is tame.
Remark 7.8. By Rosenthal's dichotomy every bounded sequence in a Banach space either has a weak Cauchy subsequence or a subsequence equivalent to the unit vector basis of l 1 (the so-called l 1 -sequence). Recall the definition of regularity of dynamical systems originally introduced by Köhler [28] for cascades in terms of independent sequences. A compact G-space X is regular if for every f ∈ C(X) the orbit f G does not contain an l 1 -sequence (in other words the second alternative is ruled out in Rosenthal's dichotomy). By Fact 3.2 it is equivalent to requiring that f G be a Rosenthal family for X for every f ∈ C(X). In fact the notions of regularity and tameness coincide (see [12] and Corollary 7.9 below). Corollary 7.9. Let X be a compact (not necessarily metrizable) G-space and f ∈ C(X). The following conditions are equivalent: (1) (G, X) is a tame dynamical system (that is, E(X) ⊂ F(X, X)).
(2) C(X) = T ame(X). Let X be a compact G-space. Then WAP functions on X come from reflexively representable factors. Similarly, Asplund functions on a compact G-system X are exactly functions which come from Asplund representable (that is, RN) factors. Every RN (being HNS) is tame in virtue of Lemma 7.4. Hence W AP (X) ⊂ Asp(X) ⊂ T ame(X) Another way to see these inclusions for metrizable X is the following proposition (see also Lemma 3.3).
Proposition 7.10. Let X be a compact metric G-space and f ∈ C(X). Then Proof. (1) Use Grothendieck's theorem: for a compact space X, a bounded subset A ⊂ C(X) is relatively weakly compact in C(X) iff it is pointwise relatively compact.
(2) By [15] we know that f ∈ Asp(X) iff f G is a fragmented family of functions on X. At the same time Lemma 3.3 shows that cls p (f G) is a (compact) metrizable subspace in R X iff f G is a fragmented family of functions on X.
Note that the equivalence "f ∈ Asp(X) ⇔ cls p (f G) is metrizable in R X " is a new characterization of Asplund functions on metric G-systems.
(1) For a concrete example of a tame system which is not RN see for instance [15,Example 14.10]. It is a cascade (X, T ) such that the subspace E(T, X) \ {T n : n ∈ Z} is homeomorphic to the "two arrows" space of Alexandroff and Urysohn (see [10, page 212], and also Ellis' example [7,Example 5.29]). It thus follows that E is a separable nonmetrizable Rosenthal compact of cardinality 2 ℵ 0 .
(2) In his paper [8] Ellis, following Furstenberg's classical work, investigates the projective action of GL(n, R) on the projective space P n−1 . It follows from his results that the corresponding enveloping semigroup is not first countable. In a later work [1], Akin studies the action of G = GL(n, R) on the sphere S n−1 and shows that here the enveloping semigroup is first countable (but not metrizable). The dynamical systems D 1 = (G, P n−1 ) and D 2 = (G, S n−1 ) are tame but not RN. Note that E(D 1 ) is Fréchet, being a continuous image of a first countable compact space, namely E(D 2 ). (3) The latter example shows that not every tame system is SUC (Strongly Uniformly Continuous) in the sense of [16]. Indeed every SUC function on a compact coset G-space is equicontinuous. (4) It is interesting to study T ame(G) for topological groups; As we know Asp(G) ⊂ T ame(G Further results concerning tame systems can be found in [28], [12], [13], [22], [27].

Banach representations of tame flows
Let us say that a compact G-space X is Rosenthal representable if it admits a faithful representation on a Rosenthal Banach space (see Definition 3.7 above).
Theorem 8.1. Every Rosenthal representable (not necessarily metrizable) compact G-space is tame. In particular, the dynamical system (Iso(V ), B * ), where B * is the weak-star compact unit ball of V * , is tame for every Rosenthal Banach space V .
Proof. It is enough to show that for every Rosenthal Banach space V the associated flow (G, X) is tame, where G = Iso (V ) is the group of linear isometries of V and X = B * the weak-star compact unit ball of V * . Now observe that E(G, X) can be identified with E := E(G, V * ) (the G-orbits in V * are relatively weak-star compact so the pointwise closure E(G, V * ) of the set of all G-translationsG = {g : V * → V * } g∈G in V * V * is compact). With this identification every element p ∈ E can be treated as a linear map p : V * → V * with norm ≤ 1. Then, for every vector f ∈ V , the composition f • p : V * → R is a linear bounded (hence norm continuous) functional on V * . That is, f • p ∈ V * * belongs to the second dual. By the reformulation of a theorem of E. Saab and P. Saab (see Fact 3.11 and Remark 3.13.1) mentioned above, the corresponding restriction f • p| X : X → R on X = B * is a fragmented function (where B * ⊂ V * is endowed with its weak-star topology). Next note that V separates points of X. Since any f • p| X is fragmented for every f ∈ V we can apply Lemma 2.3.3. It follows that p : X → X is fragmented for every p ∈ E. This means, in view of Definition 7.3, that (G, X) is tame as required.
Lemma 8.2. Let X be a compact metric G-space. The following conditions are equivalent: (1) (G, X) is representable on a separable Rosenthal Banach space.
(2) (G, X) admits countably many representations on separable Rosenthal Banach spaces which separate points of X.
Proof. Observe that the l 2 -sum of a sequence of separable Rosenthal Banach spaces is again Rosenthal. Indeed, this follows from the equivalence of (1) and (4) in Odell and Rosenthal's Theorem (Fact 3.10).
Theorem 8.3. Let X be a compact G-space, F ⊂ C(X) a Rosenthal family for X and F is G-invariant (that is, F G = F ). Then (1) there exist: a Rosenthal Banach space V , an injective mapping ν : F → V with bounded image and a continuous representation of (G, X) on V such that α is a weak * continuous map (topological embedding if F separates points of X) and Thus the following diagram commutes If X is metrizable then in addition we can suppose that V is separable and there exists a homeomorphic embedding furnishing V * * with its weak * topology, where ν 0 (K) is bounded and the following diagram commutes Let W be the symmetrized convex hull of F ; that is, Claim 1: W is also a Rosenthal family for X.
Proof. It is easy to see that F ∪ −F is a Rosenthal family for X. Now apply Proposition 4.2.
For brevity of notation let A := C(X) denote the Banach space C(X), B will denote its unit ball, and B * will denote the weak * compact unit ball of the dual space A * = C(X) * .
Claim 2: W is a Rosenthal family for B * .
Consider the sequence of sets M n := 2 n W + 2 −n B. Since W is convex and symmetric, we can apply the construction of DFJP [5] as follows. Let n be the Minkowski functional of the set M n . That is, v n = inf {λ > 0 v ∈ λM n }. Indeed, if v ∈ W then 2 n v ∈ M n , hence v n ≤ 2 −n and N (v) 2 ≤ n∈N 2 −2n < 1.
As W and B are G-invariant the natural right action V ×G → V, (v, g) → vg is isometric, that is, N (vg) = N (v). Moreover, by the definition of the norm N , we can show that this action is norm continuous (use the fact that, for each n ∈ N, the norm · n on A is equivalent to the given norm on A). Therefore, the co-homomorphism h : G → Iso (V ), h(g)(v) := vg is well defined and continuous.
Let j * : A * → V * be the adjoint map of j : V → A. Define α : X → V * as follows. For every x ∈ X ⊂ C(X) * set α(x) = j * (x). Then (h, α) is a continuous representation of (G, X) on the Banach space V . It is now easy to see that α(x), f = j(f )(x) = f (x), where f ∈ W is our original function. It follows that the function f : X → R comes from the G-system α(X) (a factor of (G, X)). We will next show that V is a Rosenthal Banach space.
Proof. The norms · n on A are equivalent to each other. It follows that if v ∈ B V then v n < 1 for all n ∈ N. That is, for every n ∈ N, v ∈ λ n M n for some 0 < λ n < 1. By the construction M n is a convex subset containing the origin. This implies that λ n M n ⊂ M n . Hence j(v) = v ∈ M n for every n ∈ N.
Claim 4: The set n∈N M n (and hence also its subset j(B V )) is sequentially precompact in the second dual (A * * , σ(A * * , A * )) (i.e. A * * endowed with its weak * topology).
Proof. We use the argument of [5, Lemma 1 (xii), p. 323] with some minor changes. For the sake of completeness we reproduce the details.
Let {c n } ∞ 1 be a sequence in k∈N M k . Then for each fixed n ∈ N and every k ∈ N we can represent c n as By Claim 2 we know that W is a Rosenthal family for B * . Thus by Fact 3.2, W is sequentially precompact in (A * * , σ(A * * , A * )). Applying a diagonal process we can choose a subsequence is σ(A * * , A * )-convergent to an element, say, x * * k ∈ A * * . In order to simplify our notation we will relabel our sequences and now assume that for every k, Claim 5: The sequence 2 k x * * k is norm Cauchy in the second dual A * * .
It is enough to find a homeomorphism η : Consider the canonical second adjoint map j * * : V * * → A * * This map is injective by [5,Lemma 1(iii)]. Using the compactness of j * * (B V ) we get j * * (B V ) = j * * (B V ) = j(B V ). Now the required homeomorphism η is the restriction of j * * to the σ(V * * , V * )-compact space B V .
Finally define ν : F → V by ν(f ) = j(f ). Clearly ν is injective being a restriction of an injective map j : V ֒→ A. It is easy to see that Clearly, ν(F ) = j(F ) is norm bounded because F is bounded and j is a bounded operator.
(2) If the compact space X is metrizable then C(X) is separable and it is also easy to see that (V, N ) is separable. Consider the injective map ν : F → V . The pointwise closure K := cls p (F ) of F in R X is a subset of bB 1 (X). Consider now the mapping T : bB 1 (X) → C(X) * * (see Remark 4.3). This map induces by Proposition 4.5 a homeomorphic embedding By construction F is a subset of V . Consider the second adjoint map Recall that this map is injective by [5,Lemma 1 (iii)]. Therefore j * * induces a homeomorphism between the compact spaces cls V * * (F ) and cls A * * (F ). Summing up we can define the desired homeomorphic embedding as follows Finally, it is easy to see that Recall again (see Subsection 6.2) that a compact space X is called Radon-Nikodým (RN) if X is homeomorphic to a weak * compact subset of the dual V * for an Asplund space V . A well known result characterizes Rosenthal spaces as those Banach spaces whose dual has the weak Radon-Nikodým property [44,. It is therefore natural to introduce the following definition.
Definition 8.4. We say that a compact topological space X is weakly Radon-Nikodým (WRN) if X is homeomorphic to a weak * compact subset of the dual V * of a Rosenthal space V . This definition agrees with the definition of WRN subsets (cf. Remark 3.13.2). More generally, a compact G-space X is called a WRN G-space if X as a G-space is Rosenthal representable.
As a direct corollary of Theorem 8.3 we get the following characterization of WRN compacta. Theorem 8.5. Let X be a compact space. The following conditions are equivalent: (1) X is WRN.
(2) There exists a Rosenthal family F ⊂ C(X) of X which separates the points of X. (3) There exists a subset F ⊂ C(X) such that F separates points of X and the pointwise closure of F in R X consists of fragmented maps from X to R.
Remark 8.6. We mention without proof that a compact G-space X is RN (that is, Asplund representable) if and only if there exists a subset F ⊂ C(X) such that F separates points of X, F is G-invariant and F is a fragmented family of functions (Definition 2.7). Comparing this with Theorem 8.10 below we see that there exists a complete analogy between RN-systems and fragmented families on one side and WRN-systems and Rosenthal families on the other.
We give here a characterization of tame functions in terms of Banach representations.
Theorem 8.7. Let X be a compact G-space. The following conditions are equivalent: (1) f : X → R is tame.
(2) f : X → R comes from a Rosenthal Banach space. That is, there exist a continuous representation (h, α) of (G, X) on a Rosenthal Banach space V and a vector v ∈ V such that If X is metrizable we can suppose in addition in (2) that V is separable. Proof.
(1) ⇒ (2): Let f ∈ T ame(X). This means By Proposition 7.7 that the orbit f G is a Rosenthal family for X. Now we can apply Theorem 8.3 to the family F := f G.
Here is the promised Banach space characterization of tame systems.
Theorem 8.8. Let X be a compact G-space. The following conditions are equivalent: (1) (G, X) is a tame G-system.
(2) ⇒ (1): Apply again Theorem 8.1 and take into account the class of tame G-systems is closed under subsystems and arbitrary products for every given G (Lemma 7.5).
(1) ⇒ (2): First of all note that C(X) = T ame(X) by Corollary 7.9. Applying Theorem 8.7 we conclude that every f ∈ C(X) = T ame(X) on a compact G-space X comes from a Rosenthal representation. Continuous functions separate points of X. This implies that there exist sufficiently many Rosenthal representations of (G, X).
We can also prove now one of our main results: Theorem 8.9. Let X be a compact metric G-space. The following conditions are equivalent: (1) (G, X) is tame.
(1) ⇒ (2): Since the compact space X is metrizable there is a sequence of functions f n ∈ C(X) = T ame(X) which separate points of X. For each f n we can construct by Theorem 8.7 a continuous Rosenthal representation (h, α) of (G, X) such that our original function f n comes from the flow (G, α(X)). Applying Lemma 8.2 we conclude that (G, X) is Rosenthal representable.
If X is a compact tame, not necessarily metrizable, dynamical G-system then the induced systems (G, B * ) (on the unit ball B * of C(X) * ) and (G, P (X)) (here P (X) denotes the compact subspace of B * consisting of all probability measures on X) are tame as well. For metrizable X this is [12,Theorem 1.5]. In fact one may show a stronger result: Theorem 8. 10. Let X be a compact G-space (no restrictions on X and G).
The following conditions are equivalent: (1) (G, X) is Rosenthal representable.
(2) There exists a G-invariant Rosenthal family A ⊂ C(X) for X which separates points of X. (1) ⇔ (3): starting with a Rosenthal family A of X we can construct a Rosenthal family for B * . Produce inductively the sequence A n := A 1 · A 1 · · · A 1 , where A 1 := A. We can suppose that A contains the constant function 1. We can show by diagonal arguments (use Fact 3.2) that A n is also Rosenthal family for X. Then it is easy to show that the G-invariant family M := ∪ n 2 −n A n is Rosenthal for X. By Proposition 4.6, M is a (G-invariant) Rosenthal family for B * . Finally observe that by Stone-Weierstrass theorem the algebra span(M ) (linear span of M in C(X)) is dense in C(X). This implies that M itself separates the points of B * . Now we apply the part (1) ⇔ (2) to the case of (G, B * ).
Remark 8.11. Let G be a topological group. One more application of Theorem 8.7 is as follows: tame functions on G are exactly matrix coefficients of continuous corepresentations of G on Rosenthal spaces. That is, a function f : G → R is tame if and only if there exist a Rosenthal space V , a strongly continuous antihomomorphism h : G → Iso(V ), vectors v ∈ V and ψ ∈ V * such that f (g) = ψ(vg) for every g ∈ G.
Indeed, choose a tame G-compactification ν : G → X of G and a continuous function F : X → R such that f = F • ν. Now we can apply Theorem 8.7 to F getting the desired V and vectors v and ψ := α(ν(e)).

8.1.
Injectivity of general tame systems and the tameness of B * . In her paper [28] Köhler also considers another useful notion, that of the enveloping operator semigroup. For a Banach space V and a linear surjective isometry T : V → V this is defined as Köhler shows that when (X, T ) is a dynamical system, V = C(X), and T : C(X) → C(X) is the operator induced by T on the space C(X), then there is always a surjective homomorphism of dynamical systems For a general dynamical system (G, X) there is an obvious analog where T g (f ) = f •g for f ∈ C(X). The following terminology was introduced in [12].
If we view P (X), the compact space of probability measures on X equipped with the weak * topology, as a subset of C(X) * with span(P (X)) = C(X) * , we see that this map Φ is nothing but the restriction of an element of E(G) to the subspace of Dirac measures {δ x : x ∈ X}.
In [12] this observation is used to give an easy proof of Köhler's theorem asserting that a tame metrizable system is injective. It is not hard to extend this statement to general tame dynamical systems. We leave the details to the reader. Theorem 8.13. Let X be a tame (not necessarily metrizable) compact Gsystem. Assume that G is a uniformly Lindelöf (e.g., second countable) group. Then (1) X is an injective system.

Compact spaces in the second dual of Rosenthal spaces
Again we remind the reader that a compact topological space K is Rosenthal if it is homeomorphic to a pointwise compact subset of the space B 1 (X) of functions of the first Baire class on a Polish space X. (1) We say that a compact space K is strongly Rosenthal if K is a subspace of B 1 (X) with compact metrizable X.
(2) We say that a compact space K is an admissible Rosenthal compactum (or simply admissible) if there exists a compact metric space X and a subset F of C(X) such that K ⊂ cls p (F ) ⊂ B 1 (X) (where as usual cls p (F ) is the pointwise closure of F in R X ).
Clearly every strongly Rosenthal compact space is Rosenthal compact, and every admissible compactum is a strongly Rosenthal compact.
Every subset F ⊂ C(X) is norm separable for a compact metric X. Hence such an F is also separable with respect to the pointwise convergence topology. Thus its pointwise closure cls p (F ) ⊂ R X is separable and therefore, in Definition 9.1, we can assume that F is countable. Furthermore the existence of a homeomorphism between R and an open interval implies that we get an equivalent definition assuming in addition that F is a uniformly bounded subset of C(X). By Definition 3.4 we have another reformulation of admissibility.
Lemma 9.2. A compact space K is admissible if and only if there exists a (countable) Rosenthal family F for a compact metric space X such that K ⊂ cls p (F ).
As mentioned in the introduction, answering a question of Talagrand [44,, R. Pol [38] gave an example of a separable compact subspace K of the space B 1 (N N ) of first Baire class functions on the irrationals such that K cannot be embedded in B 1 (X) for any compact metrizable X. This example of Pol demonstrates that not every separable Rosenthal compactum is strongly Rosenthal (and a fortiori also not admissible). We do not know whether every separable strongly Rosenthal compact space is admissible. Proof. The hereditarily property of each of these classes is obvious. In order to see that the countable product K := n K n of Rosenthal compact spaces K n is again Rosenthal we consider the topological (disjoint) sum X := n∈N X n , where X n is a Polish space for which K n ⊂ B 1 (X n ). Then K can be embedded into B 1 (X) as follows. For each element f := (f 1 , f 2 , · · · ) ∈ K n = K there exists a uniquely defined function j(f ) : X → R such that the restriction of j(f ) on X n is exactly f n . Clearly, j(f ) is a Baire 1 function on X. This defines the continuous map j : K → B 1 (X). Since j is injective and K is compact we conclude that j is a topological embedding. Suppose now that each K n is strongly Rosenthal. Then, by definition, we can assume in addition that each X n as above is a compact metric space. Now it is easy to see that K admits a topological embedding into B 1 (X * ), where X * := X ∪ {∞} is the one point compactification of X = n∈N X n . In this case we define j * : K → B 1 (X * ) by j * (f )(∞) = 0 and j * (f )(x) = j(f )(x) for every x ∈ X. Then again j * is well defined and it embeds K into B 1 (X * ).
Finally we consider the case where each K n is admissible. As in the second case we have the topological embedding j * : K → B 1 (X * ) We have to show that there exists a family F ⊂ C(X * ) such that j * (K) ⊂ cls p (F ). For each n ∈ N fix a countable subset F n ⊂ C(X n ) such that K n ⊂ cls p (F n ). It is enough to show our assertion in the case where cls p (F n ) = K n . For each k ∈ N consider the elements of the type f := (f 1 , f 2 , · · · , f k , 0 k+1 , 0 k+2 , · · · ) ∈ K n = K, where f i ∈ F i for every i ≤ k and each 0 k+m denotes the constant zero function on X k+m (again without restriction of generality we can assume that 0 t ∈ F t for every t ∈ N). Varying k ∈ N and f i ∈ F i with i ≤ k we get a countable subset F 0 ⊂ K. Clearly this subset is dense in the product space K = n∈N K n . It is easy to see that its image F := j * (F 0 ) is the required family. That is, F ⊂ C(X * ) and j * (K) ⊂ cls p (F ).
Proposition 9.4. Let X be a compact metric G-space. The following conditions are equivalent: (1) (G, X) is a tame system.
(2) ⇒ (1): If E(X) is a admissible compactum then E(X) is Rosenthal. Therefore (G, X) is a tame system by the original definition (see Theorem 7.1).
(1) ⇒ (2): Let (G, X) be a tame system. Then every continuous function f ∈ C(X) is tame. This means that f G is a Rosenthal family for X. Then the compact space E f := cls p (f G) is a subset of B 1 (X) (Proposition 7.7 and Corollary 2.6.2). Since C(X) is norm separable then f G is also norm (and hence pointwise) separable. There exists a sequence g n ∈ G such that the sequence F := {f g n } n∈N of continuous functions is pointwise dense in E f . So the compactum E f is admissible. Since X is a metrizable compact space one may choose a countable set of functions {f m } m∈N which separates the points in X. Then E(X) can be naturally embedded into the countable product K := m E fm which is admissible by Lemma 9.3.
Theorem 9.5. Let K be a compact space. The following conditions are equivalent: (1) K is a weak-star closed bounded subset in the second dual of a separable Rosenthal Banach space V . (2) K is an admissible compactum.
Proof. (1) ⇒ (2): We have to show that K is an admissible compactum. It is enough to show this for the particular case of K := B * * := B V * * of the unit ball in the second dual. Since V is separable, X := B * , the weak * compact unit ball in V * is a metrizable compact space. By our assumption V is a Rosenthal space. Then by Fact 3.10.5, K = B * * is naturally embedded into B 1 (X) with X := B * . By Goldstine's theorem the unit ball B := B V of V is weak * -dense in B * * . At the same time B can be treated as a (bounded) subset of C(X). Thus, B is a Rosenthal family for X. Hence the compactum K is admissible in the sense of Definition 9.1.2.
(2) ⇒ (1): Let K be an admissible compactum. This means that there exists a compact metric space X and a Rosenthal family F ⊂ C(X) such that K ⊂ cls p (F ) ⊂ B 1 (X). We have to show that K is homeomorphic to a weak-star closed bounded subset in the second dual of a separable Rosenthal Banach space V . It is enough to establish this for the case of K = cls p (F ). But this fact directly follows from Theorem 8.3.2.