TOPOLOGICAL TRANSFORMATION GROUPS: SELECTED TOPICS

MICHAEL MEGRELISHVILI

Abstract. We discuss some questions about continuous group actions, topological groups and representations on Banach spaces.

Contents

1. Introduction 1 2. Equivariant compactifications 2 3. Equivariant normality 5 4. Universal actions 6 5. Free topological G-groups 7 6. Banach representations of groups 8 7. Dynamical versions of Eberlein and Radon-Nikodým compacta 11 Acknowledgment 12 References 12

1. Introduction

In this paper all topological spaces are Tychonoff. A topological transformation group (ttg), or a G-space, as usual, is a triple (G, X, π), where π : G × X → X, π(g, x) := gx is a continuous action of a topological group G on a topological space X. Let G act on X_{1} and on X_{2}. A continuous map f : X_{1} → X_{2} is a G-map (or, an equivariant map) if f(gx) = gf(x) for every (g, x) ∈ G × X_{1}.

The Banach algebra of all continuous real valued bounded functions on a topological space X will be denoted by C(X). Let (G, X, π) be a G-space. It induces the action G × C(X) → C(X), with (gf)(x) = f(g^{−1}x). A function f ∈ C(X) is said to be right uniformly continuous, or also π-uniform, if the map G → C(X), g ↦ gf is norm continuous. The latter means that for every ε > 0 there exists a neighborhood V of the identity element e ∈ G such that sup_{x∈X} |f(vx)−f(x)| < ε for every v ∈ V. The set RUC_{G}(X) := RUC(X) of all right uniformly continuous functions on X is a uniformly closed G-invariant subalgebra of C(X).

A transitive action in this paper means an action with a single orbit. Let H be a closed subgroup of G and G/H be the (left) coset space endowed with the quotient topology. In the sequel we will refer to G/H as a homogeneous G-space. In this particular case f ∈ RUC_{G}(X) iff f is a uniformly continuous bounded function with respect to the natural right uniform structure on G/H (this explains Fact 2.1.1 below). A G-space X will be called:

Date: May 28, 2006.

(1) G-compactifiable, or G-Tychonoff, if X is a G-subspace of a compact Gspace.

(2) G-homogenizable, if there exists an equivariant embedding of (G, X) into a homogeneous triple (G′, G′/H) (i.e., there exists a topological group embedding h : G → G′ and a topological embedding α : X → G′/H such that h(g)α(x) = gx).

(3) G-automorphic, if X is a topological group and each translation g̃ = π(g, ·) : X → X is a group automorphism. We say also that X is a G-group.

(4) G-automorphizable, if X is a G-subspace of an automorphic G-space. In particular, if Y is a locally convex G-space with a continuous linear action of G on Y then we say that X is G-linearizable.

A G-compactification of a G-space X is a G-map ν : X → Y with a dense range into a compact G-space Y. A compactification is proper when ν is a topological embedding.

2. Equivariant compactifications

A G-compactification of a G-space X is a G-map ν : X → Y with a dense range into a compact G-space Y. A compactification is proper when ν is a topological embedding. The study of equivariant compactifications goes back to J. de Groot, R. Palais, R. Brook, J. de Vries, Yu. Smirnov and others.

The Gelfand-Raikov-Shilov’s classical functional description of compactifications admits a natural generalization for G-spaces in terms of G-subalgebras of RUC(X) (see for example [75, 11, 4]). The G-algebra V := RUC(X) defines the corresponding Gelfand (maximal ideal) space β_{G}X ⊂ V^{∗} and the, possibly improper, maximal G-compactification i_{β_{G}} : X → β_{G}X. Consider the natural homomorphism h : G → Is(RUC(X)), where Is(RUC(X) is the group of all linear isometries of RUC(X) and h(g)(f) := gf. The pair (h, i_{β_{G}}) defines a representation (in the sense of Definition 7.1) of the G-space X on the Banach space RUC_{G}(X).

A G-space is G-Tychonoff iff it can be equivariantly embedded into a compact Hausdorff G-space iff i_{β_{G}} is proper iff RUC_{G}(X) separates points and closed subsets iff (G, X) is Banach representable (cf. Definition 7.1).

Unless G is discrete, the usual maximal compactification X → βX (which always is a G_{d}-compactification ^{1} for every G-space X) fails to be a G-compactification, in general. However several standard compactifications are compatible with actions. For instance it is true for the one-point compactifications [74]. The Samuel compactification of an equiuniform G-spaces (X, μ) is a G-compactification (see [18, 74, 36]). Here ‘μ is an equiuniformity on a G-space X’ means that every translation g̃ : X → X is μ-uniform and for every entourage ε ∈ μ there exists a neighborhood U of the identity e such that (gx, x) ∈ ε for every (g, x) ∈ U × X.

1where G_{d} is the group G endowed with the discrete topology

Equiuniform precompact uniformities correspond to G-compactifications. For Gproximities see Yu. Smirnov [11]. It is easy to see that the Gromov’s compactification ^{2} of a bounded metric space (X, d) with a continuous G-invariant action is a proper G-compactification. The reason is that the function f_{z} : X → ℝ defined by f_{z}(x) := d(z, x) is π-uniform for every z ∈ X.

By J. de Vries’ well known result [76] if G is locally compact then every Tychonoff G-space is G-Tychonoff. See Palais [54] for the case of a compact Lie group G, and Antonyan [11] for compact G.

We call a group G, a V -group, if every Tychonoff G-space is G-Tychonoff. In [74], de Vries posed the ‘compactification problem’ which in our terms becomes: is every topological group G a V -group? Thus every locally compact group is a V -group. An example of [38] answers de Vries’ question negatively: there exists a ttg (G, X) such that both G and X are Polish and X is not G-Tychonoff.

Fact 2.1. Recall some useful situations when G-spaces are G-Tychonoff:

(1) Every coset G-space G/H (de Vries [74]; see also Pestov [60]);

(2) Every automorphic G-space X (and, hence, every linear G-space X), [39];

(3) Every metric G-space (X, d), where G is second category and g̃ : X → X is d-uniformly continuous for every g ∈ G, [39];

(4) Every G-space X, where X is Baire, G is uniformly Lindelöf and acts transitively on X (Uspenskij [69]).

For some results related to Fact 2.1.4 see Chatyrko and Kozlov [20].

A topological group G is uniformly Lindelöf (alternative names: ω-bounded, ℵ_{0}-bounded, ω-narrow, etc.) if for every nonempty open subset O ⊂ G countably many translates g_{n}O cover G. By a G-factorization theorem [39] every G-Tychonoff space X with uniformly Lindelöf G admits a proper G-compactification X ↪ Y with the same weight and dimension dim Y ≤ dimβ_{G}X.

The following two results are proved in [51].

(1) If G is Polish then it is a V -group iff G is locally compact.

(2) If G is uniformly Lindelöf and not locally precompact, then G is not a V - group. Furthermore there exists a Tychonoff G-space X such that i_{β_{G}} : X → β_{G}X is not injective.

The following longstanding question remains open.

Question 2.2. (Yu.M. Smirnov, 1980) Find a nontrivial Tychonoff G-space X such 1001 ? that every G-compactification of X is trivial.

The compactification problem is still open for many natural groups.

Question 2.3. [51] 1002 ? (1) Is there a locally precompact group G which is not a V-group ?

(2) What if G is the group ℚ of rational numbers ? What if G is the precompact cyclic group (ℤ, τ_{p}) endowed with the p-adic topology ?

2corresponding (closed unital) algebra is generated by the set of functions {f_{z} : X → ℝ}_{z∈X}, where f_{z}(x) := d(z, x) (see for example [4, p. 112]) .

Question 2.4. (Antonyan and Sanchis [10]) Is every locally pseudocompact group a V-group ?

Stoyanov gave (see [21, 66]) a geometric description of G-compactifications for the following natural action: X := 𝕊_{H} is the unit sphere of a Hilbert space H and G := U(H) is the unitary group endowed with the strong operator topology. Then the maximal G-compactification is equivalent to the natural inclusion of X into the weak compact unit ball 𝔹_{H} of H.

? 1003 Question 2.5. Let V be a separable reflexive Banach space. Consider a natural action of the group Is(V) on the sphere S_{V}. Is it true that the maximal Gcompactification is equivalent to the natural inclusion of X into the weak compact unit ball 𝔹_{V} of V ?

For more information about the question: ‘whether simple geometric objects can be maximal equivariant compactifications ?’ we refer to Smirnov [65].

? 1004 Question 2.6. (H. Furstenberg and T. Scarr) Let X be a Tychonoff G-space with the transitive action. Is it true that X is G-Tychonoff ?

Uspenskij’s result (see Fact 2.1.4) implies that the answer is ‘yes’ if X is Baire and G is uniformly Lindelöf.

Very little is known about the dimension of β_{G}X. Even in the case of the left regular action of G on X := G the dimension of β_{G}G (the so-called greatest ambit for G) may be greater than dim G (take a cyclic dense subgroup G of the circle group 𝕋; then dim G = 0 and dim β_{G}G = dim 𝕋 = 1). It is an old folklore result that dim β_{G}G = 0 iff G is non-Archimedean ^{3} (see for example, [56, 52]). It follows by [34, Thm 5.12] that in the case of the Euclidean group G = ℝ^{n}, we have dim β_{G}G = dim G.

? 1005 Question 2.7. Does the functor β_{G} preserve the covering dimension in case of compact Lie acting group G ?

If G is a compact Lie group then for every G-space X the inequality dim X/G ≤ dim X holds. For second countable X this is a classical result of Palais [54]. For general Tychonoff X this was done in [39] using a G-factorization theorem. This inequality does not remain true for compact (even 0-dimensional) groups. This led us [38] to an example of a locally compact Polish G-space X such that dim X = 1 and dim β_{G}X ≥ 2, where G is a 0-dimensional compact metrizable group.

Fact 2.8. [11, 9] (β_{G}X)/G = β(X/G) for every G-space X and compact G.

? 1006 Question 2.9. (Zambakhidze) Let G be a compact group, X a G-space, and B(X/G) a proper compactification of the orbit space X/G. Does there exist a proper G-compactification B_{G}(X) of X such that B_{G}(X)/G = B(X/G) ?

For some partial results see Antonyan [6] and Ageev [1].

? 1007 Question 2.10. Let G be a Polish group and X be a second countable G-Tychonoff G-space. Does there exist a metric G-completion of X with the same dimension ?

3that is, having a local base at the identity consisting of open subgroups

If G is not Polish then it is not true. The answer is affirmative if G is locally compact [40].

3. Equivariant normality

Definition 3.1. [35, 37, 51] Let (G, X, π) be a ttg.

(1) Two subsets A and B in X are π-disjoint if UA ∩ UB = ∅ for some neighborhood U of the identity e ∈ G.

(2) X is G-normal (or, equinormal) if for every pair of π-disjoint closed subsets A and B there exists a pair of π-disjoint neighborhoods O_{1}(A) and O_{2}(B). It is equivalent to say that every pair of π-disjoint closed subsets can be separated by a function from RUC_{G}(X) (Urysohn lemma for G-spaces).

(3) X is weakly G-normal if every pair of π-disjoint closed G-invariant subsets in X can be separated by a function from RUC_{G}(X).

Another version of Urysohn lemma for G-spaces appears in [29, Theorem 3.9].

Every G-normal space is G-Tychonoff. The action of G := ℚ on X := ℝ is not G-normal. One can characterize locally compact groups in terms of G-normality.

Fact 3.2. [51] For every topological group G the following are equivalent:

(1) Every normal G-space is G-normal.

(2) G is locally compact.

It is unclear if ‘G-normal’ can be replaced by ‘weakly G-normal’.

Question 3.3. Is every second countable G-space weakly G-normal for the group 1008 ? G := ℚ of rational numbers ?

If not, then by [51, Theorem 3.2] one can construct for G := ℚ a Tychonoff Gspace X which is not G-Tychonoff. That is, it will follow that ℚ is not a V-group (see Question 2.3).

Fact 3.4. Every coset G-space G/H is G-normal.

Then the following ‘concrete’ actions (being coset spaces) are equinormal :

(1) (U(H), 𝕊_{H}) for every Hilbert space H;

(2) (Is(𝕌), 𝕌) (where Is(𝕌) is the isometry group of the Urysohn space 𝕌 with the pointwise topology);

(3) (GL(V), V \ {0}) for every normed space V (see [43]).

(4) (GL(V), ℙ_{V}) for every normed space V and its projective space ℙ_{V}.

It follows in particular by (4) that ℙ_{V} is GL(V)-Tychonoff. The latter is well known among experts and easy to prove (cf. e.g. Pestov [58]) using equiuniformities.

Question 3.5. Is it true that the following (G-Tychonoff) actions are G-normal: 1009 ? (U(l_{2}), l_{2}), (Is(l_{p}), 𝕊_{l_{p}})), p > 1 (p ≠ 2) ?

4. Universal actions

Let A be some class of continuous actions (G, X). We say that a pair (G_{u}, X_{u}) from A is (equivariantly) universal for the class A if for every (G, X) ∈ A there exists an equivariant ^{4} pair (h, f) such that h : G ↪ G_{u} is a topological group embedding and f : X ↪ X_{u} is a topological embedding. If, in addition we require that G = G_{u} and h = id_{G} then we simply say that X_{u} is G-universal.

For a compact space X denote by H(X) the group of all homeomorphisms of X endowed with the compact open topology.

Fact 4.1. (1) (Antonyan and de Vries [12]; Tychonoff theorem for G-spaces) For every locally compact sigma-compact group G and a cardinal ^{5} τ there exists a universal G-space of weight τ.

(2) (Megrelishvili [39]; G-space version of Nagata’s universal space theorem) Let G be a locally compact sigma-compact group of weight w(G) ≤ τ. For every integer n ≥ 0 there exists, in the class of metrizable G-spaces of dimension ≤ n and weight ≤ τ, a universal G-space.

(3) (Hjorth [32]) If G is a Polish group, then the class of Polish G-spaces has a G-universal object.

(4) (Megrelishvili and Scarr [52]; Equivariant universality of the Cantor cube) Let K := {0, 1}^{ℵ_{0}} be the Cantor cube. Then (H(K), K) is equivariantly universal for the class of all 0-dimensional compact metrizable G-spaces, where G is second countable and non-Archimedean.

See also results of Becker and Kechris [14, Section 2.2.6], Vlasov [73] and questions posed by Iliadis in [33, p. 502].

? 1010 Question 4.2. Let G be a Polish group. Is it true that there exists a universal G-space in the class of all Polish G-spaces with dimension ≤ n ?

Fact 4.3. [42]

(1) (H(I^{ℵ_{0}}), I^{ℵ_{0}}) is equivariantly universal for the class of all G-compactifiable actions (G, X) with second countable G and X.

(2) Let G be a uniformly Lindelöf group. Then every G-Tychonoff space X is equivariantly embedded into (H(I^{τ}), I^{τ}) where τ ≤ w(X)w(G).

As a trivial corollary of Fact 4.3.1 is a well known result of Uspenskij [68] about universality of the group H(I^{ℵ_{0}}). Another proof of Uspenskij’s result (see [7, Corollary 4]) follows by the following Theorem of Antonyan.

Fact 4.4. [7] Let G be a uniformly Lindelöf group. Then for every G-Tychonoff space X there exists a family of convex metrizable G-compacta {K_{f}}_{f∈F} such that |F | = w(X) and X possesses a G-embedding into the product ∏_{f∈F} K_{f}.

The following natural question of Antonyan remains open (even for τ = ℵ_{0}).

? 1011 Question 4.5. (Antonyan [7, 8]) Let G be a uniformly Lindelöf group of weight wG ≤ τ. Does there exist a G-universal compact G-space of weight τ ?

? 1012 Question 4.6. Let τ be an uncountable cardinal.

4As usual, equivariant means that f(h(g)x) = gf(x)

5All cardinals in this paper are infinite.

(1) Is it true that there exists an equivariantly universal ttg (G_{u}, X_{u}) in the class of all ttg’s (G, X) where X is G-Tychonoff and max{w(G), w(X)} ≤ τ ?

(2) What if G_{u} and G are abelian ?

Positive answer on (1) will imply the solution of the following question.

Question 4.7. (Uspenskij [71]) Does there exist a universal topological group of 1013 ? every given infinite weight τ?

Fact 4.8. G-Compactifiable ⊃ G-Homogenizable ⊃ G-Automorphizable

Every G-group X is naturally identified with the coset P -space P/G, where P := X ⋋ G is the corresponding semidirect product. This explains the second inclusion. The first inclusion follows by Fact 3.4.

If G is locally compact then every G-space is G-linearizable (see for example, [11, 77]) and all classes from Fact 4.8 coincide.

It is well known that the action of H(I^{ℵ_{0}}) on I^{ℵ_{0}} is transitive. Using Effros’ theorem one can show that Tychonoff cubes I^{λ} are coset H(I^{λ})-spaces for every infinite power, [42]. Therefore Fact 4.3 leads to the equality G-Compactifiable = G-Homogenizable for every uniformly Lindelöf group G. It is unclear in general.

Question 4.9. Is there a G-Tychonoff non-homogenizable G-space ? Equivalently, 1014 ? is every compact G-space G-homogenizable?

5. Free topological G-groups

Let X be a Tychonoff G-space. By F_{G}(X) we denote the corresponding free topological G-group in the sense of [42]. Recall a link with the epimorphism problem. Uspenskij has shown in [70] that in the category of Hausdorff topological groups epimorphisms need not have a dense range. This answers a longstanding problem by K. Hofmann. Pestov gave [55, 57] a useful epimorphism criteria in terms of the free topological G-groups.

Fact 5.1. (Pestov [55]) The natural inclusion H ↪ G of a topological subgroup H into G is an epimorphism (in the category of Hausdorff groups) iff the free topological G-group F_{G}(X) of the coset G-space X := G/H is trivial ^{6}.

For instance by results of [42], F_{G}(X) is trivial in the following situation: the group G := H(𝕊) is the group of all homeomorphisms of the circle 𝕊 which can be identified with the compact coset G-space G/St(z) (where z is a point of 𝕊 and St(z) is the stabilizer of z). It follows that St(z) ↪ G is an epimorphism. This example shows also that not every compact G-space is G-automorphizable.

If G is locally compact then F_{G}(X) canonically can be identified with the usual free topological group F (X). This suggests the following questions.

Question 5.2. Let X be G-automorphic (i.e., the canonical map X → F_{G}(X) is an 1015 ? embedding). Is it true that the natural map F (X) → F_{G}(X) is a homeomorphism ?

Question 5.3. Let X be a G-automorphic G-space. Is it true that F_{G}(X) is alge- 1016 ? braically free over X ?

6Triviality means, ‘as trivial as possible’, isomorphic to the cyclic discrete group.

6. Banach representations of groups

A representation of a topological group G on a Banach space V is a homomorphism h : G → Is(V), where Is(V) is the topological group of all linear surjective isometries V → V endowed with the strong operator topology inherited from V^{V}. Denote by V_{w} the space V in its weak topology. The corresponding topology on Is(V) inherited from V_{w}^{V} is the weak operator topology. By [45], for a wide class PCP (Point of Continuity Property) of Banach spaces, including reflexive spaces, strong and weak operator topologies on Is(V) coincide.

Let K be a ‘well behaved’ subclass of the class Ban of all Banach spaces. Important particular cases for such K are: Hilb, Ref or Asp, the classes of Hilbert, reflexive or Asplund spaces respectively. The investigation of Asp and the closely related Radon-Nikodým property is among the main themes in Banach space theory. Recall that a Banach space V is an Asplund space if the dual of every separable linear 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. Namioka’s Joint continuity theorem implies that every weakly compact set in a Banach space is norm fragmented. This explains why every reflexive space is Asplund. For more details cf. [53, 17, 24]. For some applications of the fragmentability concept for ttg’s see [44, 48, 45, 27].

We say that a topological group G is K-representable if there exists a representation h : G → Is(V) for some V ∈ K such that h is a topological embedding; notation: G ∈ K_{r}. In the opposite direction, we say that G is K-trivial if every continuous K-representation of G is trivial. Of course, T opGr = Ban_{r} ⊃ Asp_{r} ⊃ Ref_{r} ⊃ Hilb_{r}. As to T opGr = Ban_{r}, it is an old observation due to Teleman [67] (see also [57]) that for every topological group G the natural representation G → Is(V) on the Banach space V := RUC(G) is an embedding.

Every locally compact group is Hilbert representable^{7} (Gelfand-Raikov). On the other hand, even for Polish groups very little is known about their representabilty in well behaved Banach spaces.

It is also well known that T opGr ≠ Hilb_{r}. Moreover, there are examples of unitarily trivial, so-called exotic groups (Herer-Christensen [31] and Banasczyk [13]).

Classical results imply that a group is unitarily representable iff the positive definite functions separate the closed subsets and the neutral element. By results of Shoenberg the function f(v) = e^{−∥x∥^{p}} is positive definite on L_{p}(μ) spaces for every 1 ≤ p ≤ 2. An arbitrary Banach space V, as a topological group, cannot be exotic because the group V in the weak topology is unitarily representable. However C[0, 1], c_{0} ∉ Hilb_{r} (see Fact 6.6 below).

Fact 6.1. [46] A topological group G is (strongly) reflexively representable (i.e., G is embedded into Is(V) endowed with the strong operator topology for some reflexive V) iff the algebra W AP (G) of all weakly almost periodic functions determines the topology of G.

7we say also: unitarily representable

A weaker result replacing ‘strong’ by ‘weak’ appears earlier in Shtern [64]. The group G := H_{+}[0, 1] of orientation preserving homeomorphisms of the closed interval (with the compact open topology) is an important source for counterexamples.

Fact 6.2. (1) [46] H_{+}[0, 1] is reflexively (and hence also Hilbert) trivial. (2) [28] Moreover, H_{+}[0, 1] is even Asplund trivial.

The question if W AP (G) determines the topology of a topological group G was raised by Ruppert [63]. (1) means that every wap function on H_{+}[0, 1] is constant. The WAP triviality of G := H_{+}[0, 1] was conjectured by Pestov.

Question 6.3. (Glasner and Megrelishvili) Is there an abelian group which is not 1017 ? reflexively representable ?

Equivalently: is it true that the algebra W AP (G) on an abelian group G separates the identity from closed subsets ?

Question 6.4. Is it true that every Banach space X, as a topological group, is 1018 ? reflexively representable ?

A separable Banach space U is uniformly universal if every separable Banach space, as a uniform space, can be embedded into U. Clearly, C[0, 1] is linearly universal and hence also uniformly universal. In [2] Aharoni proved that c_{0} is uniformly universal. P. Enflo [23], in answer to a question by Yu. Smirnov, found in 1969 a countable metrizable uniform space which is not uniformly embedded into a Hilbert space. That is, l_{2} is not uniformly universal ^{8}. However, it is not clear if ‘Hilbert’ may be replaced by ‘reflexive’.

Question 6.5. Does there exist a uniformly universal reflexive Banach space ? 1019 ?

There is no linearly universal separable reflexive Banach space (Szlenk). Moreover, there is no Lipschitz embedding of c_{0} into a reflexive Banach space (Mankiewicz). For more information on uniform classification of Banach spaces we refer to [15].

Fact 6.6. [41, 45] Let G be a (separable) metrizable group and let U_{L} denote its left uniform structure. If G is reflexively representable, than (G, U_{L}) as a uniform space is embedded into a (separable) reflexive space V. Moreover, if G is unitarily representable then G is uniformly embedded into a (separable) Hilbert space.

As a corollary it follows that C[0, 1] and c_{0} are not unitarily representable. Positive answer to the following question will imply the positive answer on 6.5.

Question 6.7. Are the additive groups c_{0} and C[0, 1] reflexively representable ? 1020 ?

A natural question arises about coincidence of Ref_{r} and Hilb_{r}. The positive answer was conjectured by A. Shtern [64]. By [47], L_{4}[0, 1] ∈ Ref_{r} and L_{4}[0, 1] ∉ Hilb_{r}. Chaatit [19] proved that every separable L_{p}(μ) space (1 ≤ p < ∞), is reflexively representable.

8This result by Enflo has recently led to some exciting developments in geometric group theory, cf. Gromov [30].

By [3], if a metrizable abelian ^{9} group, as a uniform space, is embedded into a Hilbert space then positive definite functions separate the identity and closed subsets. Combining this with Fact 6.6 we have the following ^{10}.

Fact 6.8. A metric abelian group is unitarily representable iff it is uniformly embedded into a Hilbert space.

The same observation (for second countable abelian groups) is mentioned by J. Galindo in a recent preprint [25]. Facts 6.6 and 6.8 suggest the following question.

? 1021 Question 6.9. (See also [47]) Let G be a metric group and it, as a uniform space (G, U_{L}), is uniformly embedded into a reflexive (Hilbert) Banach space. Is it true that G is reflexively (resp., unitarily) representable ?

Galindo announced [25] that for every compact space X the free abelian topological group A(X) is unitarily representable. Uspenskij found [72] that in fact this is true for every Tychonoff space X. The case of F (X) is open.

? 1022 Question 6.10. Let X be a Tychonoff (or, even a compact) space.

(1) Is it true that the free topological group F (X) is reflexively representable ?

(2) (see also Pestov [61]) Is it true that F (X) is unitarily representable ?

U(l_{2}) clearly is universal for Polish unitarily representable groups.

? 1023 Question 6.11. Does there exist a universal reflexively representable Polish group ?

? 1024 Question 6.12. Is it true that if G is reflexively representable then the factor group G/H is also reflexively representable ?

It is impossible here to replace ‘reflexively’ by ‘Hilbert’ because every Abelian Polish group is a factor-group of a Hilbert representable Polish group (Gao and Pestov [26]). A positive answer to Question 6.12 will imply that every second countable Abelian group is reflexively representable. Also then we will get a negative answer to the following problem.

? 1025 Question 6.13. (A.S. Kechris) Is every Polish (non-Abelian) topological group a topological factor-group of a subgroup of U(ℓ_{2}) with the strong operator topology ?

A natural test case by Fact 6.2 is the group H_{+}[0, 1]. Fact 6.2 of course implies that every bigger group G ⊃ H_{+}[0, 1] is not reflexively representable. Moreover if G in addition is topologically simple then it is reflexively trivial. For instance the Polish group Is(𝕌_{1}) ^{11} is reflexively trivial (as observed by Pestov [59], this fact follows immediately from results by Megrelishvili [46] and Uspenskij [71]). It follows that every Polish group is a subgroup of a reflexively trivial group.

? 1026 Question 6.14. (Glasner and Megrelishvili [28]) Is it true that there exists a nontrivial Polish group which is reflexively (Asplund) trivial but does not contain a subgroup topologically isomorphic to H_{+}[0, 1] ?

9in fact, metrizable amenable, is enough.

10It was presented on Yaki Sternfeld Memorial Conference (Israel, May 2002).

11where 𝕌_{1} is a sphere of radius 1/2 in 𝕌.

By a recent result of Rosendal and Solecki [62] every homomorphism of H_{+}[0, 1] into a separable group is continuous. Hence every representation (of a discrete group) H_{+}[0, 1] on a separable reflexive space is trivial.

Question 6.15. (Glasner and Megrelishvili) Find a Polish group G which is reflex- 1027 ? ively (Asplund) trivial but the discrete group G_{d} admits a nontrivial representation on a separable reflexive (Asplund) space.

Question 6.16. Is it true that the group H(I^{ℵ_{0}}) is reflexively trivial ? 1028 ?

It is enough to show that the group H(I^{ℵ_{0}}) is topologically simple.

Question 6.17. (Glasner and Megrelishvili) Is it true that there exists a group G 1029 ? such that G ∈ Asp_{r} and G ∉ Ref_{r}.

7. Dynamical versions of Eberlein and Radon-Nikodým compacta

Eberlein compacta in the sense of Amir and Lindenstrauss [5] are exactly the weakly compact subsets in Banach (equivalently, reflexive) spaces V. If X is a weak^{∗} compact subset in the dual V^{∗} of an Asplund space V then, following Namioka [53], X is called Radon-Nikodým compact (in short: RN). Every reflexive Banach space is Asplund. Hence, every Eberlein compact is RN.

Definition 7.1. [48] A (proper) representation of (G, X) on a Banach space V is a pair (h, α) where h : G → Is(V) is a continuous homomorphism of topological groups and α : X → V^{∗} is a weak star continuous bounded G-mapping (resp. embedding) with respect to the dual action G×V^{∗} → V^{∗}, (gφ)(v) := φ(h(g^{−1})(v)).

Note that the dual action is norm continuous whenever V is an Asplund space, [44]. It is well known that the latter does not remain true in general.

A G-space X is properly representable on some Banach space V if and only if X is G-Tychonoff (consider the natural representation on V := RUC_{G}(X)).

The following dynamical versions of Eberlein and Radon-Nikodým compact spaces were introduced in [48]. A compact G-space X is Radon-Nikodým, RN for short, if there exists a proper representation of (G, X) on an Asplund Banach space V. If V is reflexive (Hilbert) then we get the definitions of reflexively (resp., Hilbert) representable G-spaces. In the first case we say that (G, X) is an Eberlein G-space.

Fact 7.2. Let X be a metric compact G-space.

(1) [48] X, as a G-space, is Eberlein (i.e., reflexively G-representable) iff X is a weakly almost periodic G-space in the sense of Ellis [22].

(2) [27] X, as a G-space, is RN iff X is hereditarily nonsensitive.

Compact spaces which are not Eberlein are necessarily non-metrizable, while even for G := ℤ, there are natural metric compact G-spaces which are not RN.

There exist compact metric ℤ-spaces which are reflexively but not Hilbert representable [49]. This answers a question of T. Downarowicz.

Question 7.3. Is it true that Eberlein (that is, reflexively representable) compact 1030 ? G-spaces are closed under factors ?

For the trivial group G (i.e., in the purely topological setting) the answer is affirmative and this is just a well known result by Benyamini, Rudin and Wage [16]. The answer is ‘yes’ for compact metric G-spaces.

? 1031 Question 7.4. Is it true that RN (that is, Asplund representable) compact G-spaces are closed under factors ?

For the trivial group one can recognize a longstanding open question by Namioka [53]. Again if X is metric then the answer is ‘yes’. For Hilbert representable actions the situation is unclear even for the metric case.

? 1032 Question 7.5. Is it true that Hilbert representable compact metric G-spaces are closed under factors ?

For a compact G-space X denote by E := E(X) the corresponding (frequently, ‘huge’) compact right topological (Ellis) enveloping semigroup. It is the pointwise closure of the set of translations {g̃ : X → X}_{g∈G} in the product space X^{X}.

The enveloping semigroup E(X) of a metric compact RN G-space X is a separable Rosenthal compact (hence, card(E(X)) ≤ 2^{ℵ_{0}}), [27].

? 1033 Question 7.6. (Glasner and Megrelishvili) Is it true that for every compact metric RN G-space X the enveloping semigroup E(X) is metrizable ? ^{12}

A function f ∈ RUC(G) is Asplund, notation: f ∈ Asp(G), if f is a (generalized) matrix coefficient of an Asplund representation h : G → Is(V). This means that V is Asplund and there exists a pair of vectors (v, ψ) ∈ V × V^{∗} such that f(g) = ψ(g^{−1}v). Similarly, W AP (G) is the set of all matrix coefficients of reflexive representations. Recall that if RUC(G) = W AP (G) then G is precompact [50].

? 1034 Question 7.7. Assume that RUC(G) = Asp(G). Is it true that G is precompact ?

Acknowledgment

I am indebted to Yu. Smirnov who led me to the world of topological transformation groups. I thank S. Antonyan, D. Dikranjan, E. Glasner, V. Pestov and V. Uspenskij for helpful suggestions.

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Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel

   E-mail address: megereli@math.biu.ac.il
   URL: http://www.math.biu.ac.il/^{∼}megereli