07.04.98

Eberlein Groups

by Michael G. Megrelishvili Department of Mathematics and Computer Science Bar-Ilan University Ramat-Gan, 52900, Israel

Abstract. We show that the algebra W (G) of all weakly almost periodic functions generates the given topology on a Hausdorff topological group G (say, Eberlein group) if and only if there exists a reflexive Banach space X such that G is a topological subgroup of the group Is(X) of all linear isometries of X endowed with the strong operator topology. For this purpose we establish a general result on coincidence of the strong and weak operator topologies on Is(X) for arbitrary reflexive X. The proof is based on the Ellis-Lawson joint continuity theorem and our recent result about continuity of dual group actions on Asplund spaces. In the last part we discuss two possible ways finding non-Eberlein groups using uniformly universal Banach spaces or minimal topological groups.

§1. Introduction.

Let S be a semitopological semigroup, that is, a topologized semigroup with a separately continuous multiplication. We will denote by C(S) the commutative C^{∗}- algebra of all bounded continuous complex valued functions on S. For each s ∈ S, the right translation maps R_{s} of C(S) into itself are defined by

R_{s}f(x) = f(xs) for all x ∈ S.

Recall some basic facts about weak almost periodicity (see [Eb], [LG], [BJM], [R1]). A function f ∈ C(S) is weakly almost periodic if the orbit of f, that is, the set

Sf := {R_{s}f | s ∈ S}

is relatively weakly compact in C(S). The set W (S) of all such functions is a closed S-invariant subalgebra of C(S). If S is compact, then W (S) = C(S). Moreover, the compactification u: S → S^{w}, induced by the algebra W (S), is the universal semitopological semigroup compactification of S.

Typeset by AMS-TEX

AMS Subject Classification: 54H15, 22A20, 43A65, 46B10

Key words and phrases. Compact semitopological semigroup, weakly almost periodic, Ellis-Lawson theorem.

If (X, ∥ ∥) is a Banach space, we denote by (L(X), ∥ ∥) the algebra of all bounded linear operators X → X. The dual Banach space of X will be denoted by X^{∗}. We use the following notation:

B(X) := {x ∈ X | ∥x∥ ≤ 1}
Cont(X) := {s ∈ L(X) | ∥s∥ ≤ 1}.

The group of all linear isometries of X will be denoted by Is(X). The strong, strong^{∗} and weak operator topology (which we denote respectively by T_{s}, T_{s^{∗}} and T_{w}) on L(X) is the weakest topology generated respectively by the system of maps:

(1) {x̃: L(X) → X, x̃(s) = sx | x ∈ X},

(2) {f̃: L(X) → X^{∗}, f̃(s)(x) = f(sx) | f ∈ X^{∗}},

(3) {ψ_{x,f}: L(X) → ℝ, ψ_{x,f}(s) = f(sx) | x ∈ X, f ∈ X^{∗}}.

If a subset P of L(X) is endowed with one of the following subspace topologies T_{s} |_{P} , T_{s^{∗}} |_{P} , T_{w} |_{P} , then often we indicate this by writing P_{s}, P_{s^{∗}} and P_{w}, respectively. Analogously, a subset A of X endowed with its usual weak topology is denoted by A_{w}. L(X)_{s}, L(X)_{s^{∗}}, L(X)_{w} are always Hausdorff semitopological semigroups.

Fact 1.1. (Banach-Bourbaki Theorem) A Banach space X is reflexive iff B(X) is weakly compact.

Fact 1.2. The semitopological semigroup Cont(X)_{w} is compact iff X is reflexive.

Proof. The compactness of Cont(X)_{w} for a reflexive X is well-known (see, for instance [LG, Th. 3.1]). The converse implication follows from Fact 1.1, taking into account that for any fixed vector x_{0} with ∥x_{0}∥ = 1, the map

Cont(X)_{w} → B(X)_{w}, s → sx_{0}

is continuous and onto. Indeed, take a continuous functional f on X such that f(x_{0}) = 1 and ∥f∥ = 1. For every z ∈ B(X) assign to the pair (f, z) the following linear operator

A_{f,z}: X → X, A_{f,z}(x) = f(x)z

Clearly, A_{f,z} is a contraction of X moving x_{0} into z. □

The following result is well known for Hilbert spaces.

Theorem 1.3. Let X be a reflexive Banach space. Then the strong and weak operator topologies coincide on Is(X).

Proof. In [Me1, Corollary 6.9] we have already proved that the reflexivity of X guarantees the continuity of the dual action of Is(X) on X^{∗}. This leads (see [Me1, Corollary 6.11] to the equality T_{s} |_{Is(X)}= T_{s^{∗}} |_{Is(X)} . Therefore, for our purposes it suffices to show that T_{s^{∗}} |_{Is(X)}⊆ T_{w} |_{Is(X)} .

Consider the canonical separately continuous semigroup action

α: Cont(X)_{w} × B(X)_{w} → B(X)_{w}.

The reflexivity of X guarantees that B(X)_{w} and Cont(X)_{w} are compact. Therefore we can apply the Ellis-Lawson Theorem [La, Corollary 5.2] which implies that α is jointly continuous at each point (g, x), where g is an arbitrary unit of Cont(X) and x is an arbitrary point of B(X). Thus, the restricted group action

α′: Is(X)_{w} × B(X)_{w} → B(X)_{w}

is jointly continuous. Let f ∈ X^{∗}. Clearly, the restricted map f |_{B(X)} : B(X)_{w} → ℂ is continuous. Using the joint continuity of α′ and the compactness of B(X)_{w}, we obtain that for a given ε > 0 and g_{0} ∈ Is(X) there exists a neighborhood O(g_{0}) of g_{0} in Is(E)_{w} such that

|f(gx) − f (g_{0}x)| < ε

for every x ∈ B(X) and g ∈ O(g_{0}). Or, equivalently,

∥fg − fg_{0}∥^{∗} < ε

for every g ∈ O(g_{0}). This means that the orbit map

f̃: Is(X)_{w} → X^{∗}

is continuous for every f ∈ X^{∗}. Therefore, T_{s^{∗}} |_{Is(X)}⊆ T_{w} |_{Is(X)}, as required. □

Remark 1.4. In general, T_{s} |_{Is(X)}≠ T_{w} |_{Is(X)}. The following general construction, providing many counterexamples, is based on an idea of Helmer [H, Ex. 13]. Let Y be a compact Hausdorff space and let G be a subgroup of the group H(Y) of all autohomeomorphisms of Y. Denote by G_{p} the group G endowed with the topology of pointwise convergence. Then the evaluation map α: G_{p} × Y → Y is separately continuous. Consider the induced action

π: G × C(Y) → C(Y), (g ∘ f)(y) = f(g^{−1}y)

and the induced injective group homomorphism j: G → Is(X), where X = C(Y). Now, suppose that α is not jointly continuous and G is a k-space (for example, in [H, Ex. 13], G is the topological group of all rationals and Y is the square [−1, 1]^{2}). Then, by Grothendieck’s classical result [G, Th. 5], the map j: G_{p} → Is(X)_{w} is compact-preserving and, hence, continuous by our assumption on G. It is easy to show that T_{w} |_{j(G)}≠ T_{s} |_{j(G)} . Indeed, assuming the contrary, we will obtain that the restricted dual action

π_{B}^{∗}: G_{p} × B(X^{∗})_{w^{∗}} → B(X^{∗})_{w^{∗}} , (g ∘ ψ)(f) = ψ(g^{−1} ∘ f)

is continuous, where B(X^{∗})_{w^{∗}} is the unit ball of X^{∗} endowed with the weak^{*} topology. Then the original action α (being canonically equivalent to a subaction of π_{B}^{∗}) is jointly continuous. This contradicts our assumption.

Remark 1.5. In general, T_{s} |_{Is(X)}≠ T_{s^{∗}} |_{Is(X)} . Indeed, it is well known that there are many continuous norm invariant continuous linear group actions on Banach spaces X such that the corresponding dual actions on X^{∗} are not continuous. Consider, for example, X = ℓ_{1} and define a subgroup S(ℕ) of Is(X)_{s} consisting of all permutations of “coordinates.” Then the dual action of S(ℕ) on ℓ_{1}^{∗} = m is not continuous. Hence, it is not even true that T_{s^{∗}} |_{Is(X)}⊆ T_{s} |_{Is(X)} . For arguments in the case of X = C[0, 1], see [Me1]. Note also that if X is Asplund (by Stegall’s result [St], it is equivalent to saying that the dual X^{∗} has the Radon-Nikodym property), then, necessarily, T_{s^{∗}} |_{Is(X)}⊆ T_{s} |_{Is(X)} (cf. [Me1, Corollary 6.9]).

§2. Compact semitopological semigroups “live” in reflexive spaces.

For every reflexive Banach space X, the semitopological semigroup Cont(X)_{w} is compact. The aim of this section is to show

Theorem 2.1. Let (S, τ) be a compact Hausdorff semitopological semigroup. Then there exists a reflexive Banach space X such that (S, τ) is a subsemigroup of the compact semitopological semigroup Cont(X)_{w}.

Proof. Without loss of generality, we can assume that S has the identity e. Consider the natural monoid action

α: S × C(S) → C(S) α(s, f) = sf = R_{s}(f).

Clearly, each s-translation α^{s} = R_{s} is continuous. Moreover, by [G, Th. 5], each orbit map

α_{f}: S → C(S) , s ↦ sf

is weakly continuous (see [LG]). Therefore, for every fixed f ∈ C(S), the orbit Sf is weakly compact. Denote by E_{f} the Banach subspace of C(S) linearly and topologically generated by Sf. Since a^{s} is continuous for every s ∈ S, E_{f} is Sinvariant. By the Hahn-Banach Theorem , the weak topology of E_{f} is the same as its relative weak topology as a subset of C(S). In particular, Sf is weakly compact in E_{f}. By the Krein-Smulian Theorem, the convex hull co(−Sf ∪ Sf) = W of the weakly compact symmetric subset −Sf ∪ Sf is relatively weakly compact. That is, the (weak) closure of W in E_{f} is weakly compact. Since W is a convex, bounded and symmetric subset of E_{f}, we can apply the factorization procedure discovered by Davis, Figiel, Johnson and Pelczynski [DFJP]. For each natural n, set

U_{n} = 2^{n}W + 2^{−n}B(E_{f}).

Let ∥ ∥_{n} be the gauge of the set U_{n}. That is,

∥x∥_{n} = inf {λ > 0 | x ∈ λU_{n}}.

Then, using [DFJP, §2] we obtain:

(1) ∥ ∥_{n} is a norm on E_{f} equivalent to the given norm ∥ ∥ of E_{f};

(2) For x ∈ E_{f}, let

N(x) = (∑_{n=1}^{∞} ∥x∥_{n}^{2})^{1/2} ,

and let

X_{f} = {x ∈ E_{f} | N(x) < ∞}.

Denote by j: X_{f} → E_{f} the inclusion map;

(3) f ∈ Sf ⊆ W ⊆ B(X_{f});

(4) (X_{f}, N) is a Banach space and j: X_{f} → E_{f} is a continuous linear injection;

(5) X_{f} is reflexive;

(6) The restriction of j: X_{f} → E_{f} on each bounded subset A of X_{f} induces a homeomorphism of A and j(A) in the weak topologies.

Proof. Consider the weak closure cl_{w}(A) of A in X_{f}. By the reflexivity of X_{f}, the set cl_{w}(A) is weakly compact. Hence, j, being weakly continuous and injective, induces a homeomorphism of cl_{w}(A) and j(cl_{w}(A)) with respect to the weak topologies. This proves assertion (6).

(7) N(sx) ≤ N(x) for every x ∈ X_{f} and every s ∈ S.

Proof. It suffices to show that ∥sx∥_{n} ≤ ∥x∥_{n} for every n ∈ ℕ. By our construction sW ⊆ W and sB(E_{f}) ⊆ B(E_{f}) (R_{s} is a contraction of E_{f}). Then, from x ∈ λ(2^{n}W + 2^{−n}B(E_{f})) we obtain that sx ∈ λ(2^{n}(sW) + 2^{−n}s(B(E_{f}))) ⊆ λ(2^{n}W + 2^{−n}B(E_{f})). Hence, ∥sx∥_{n} ≤ ∥x∥_{n}, as required. This proves assertion (7).

As a corollary, we get that X_{f} is an S-invariant subset of E_{f}. Therefore, the restricted action

α_{f}: S × X_{f} → X_{f}

is well-defined.

(8) For every z ∈ X_{f}, the orbit map

z̃: S → X_{f} ,  z̃(s) = sz

is weakly continuous.

Proof. Indeed, by assertion (7), the orbit z̃(S) = Sz is an N-normed bounded subset in X_{f}. Our assertion follows from (6) (for A = Sz), taking into account that z̃: S → E_{f} is weakly continuous.

By (7), for every s ∈ S, the translation map α_{f}^{s}: X_{f} → X_{f} is a linear contraction of (X_{f}, N). Therefore, we get the map

γ_{f}: S → Cont(X_{f}) , γ_{f}(s) = α_{f}^{s}.

Now, directly from (8) we obtain (9) γ_{f}: S → Cont(X_{f})_{w} is a continuous monoid homomorphism.

Now we are ready to construct the desired reflexive Banach space X. Consider the family F = {X_{f} | f ∈ C(S)} of reflexive Banach spaces and the family

{γ_{f}: S → Cont(X_{f}) | f ∈ C(S)}

of monoid homomorphisms . Define X as the ℓ_{2}-product (cf. [Da, p. 35] or [NP, p. 743]), X = ∏_{2} X_{f} of the family F. Recall that it is the space of all functions x = (x_{f}) such that x_{f} ∈ X_{f} for each f ∈ C(S), and the norm on X is defined by

∥x∥ = ( ∑_{f∈C(S)} ∥x_{f}∥^{2})^{1/2} < ∞.

Then (X, ∥ ∥) is reflexive. Moreover, (∏_{2} X_{f})^{∗} = ∏_{2} X_{f}^{∗} and the corresponding pairing for x = (x_{f}) ∈ ∏_{2} X_{f}, h = (h_{f}) ∈ ∏_{2} X_{f}^{∗} is defined by

h(x) = ∑_{f∈C(S)} h_{f}(x_{f}).

Now we define a linear representation of S in X as the ℓ_{2}-product of old representations. Precisely, we define

γ: S → Cont(X) , γ(s)(x_{f}) = (sx_{f}).

First observe that by assertion (7), X is well-defined. Clearly, γ is a monoid homomorphism. By assertion (9) and the above-mentioned description of X^{∗}, it is easy to show that γ is weakly continuous. In order to establish that γ is the desired embedding, by the compactness of S, we have only to show that γ is injective. Equivalently, it suffices to check that {γ_{f} | f ∈ C(S)} separates the points of S. Let s_{1}, s_{2} be distinct points of S. Choose a continuous function f ∈ C(S) with f(s_{1}) ≠ f(s_{2}). Since (s_{1}f)(e) = f(s_{1}) and (s_{2}f)(e) = f(s_{2}), it follows that s_{1}f and s_{2}f are distinct elements of C(S) and of E_{f}. Moreover, by our construction, X_{f} ⊆ E_{f} and s_{1}f, s_{2}f both belong to X_{f} (see assertion (3)). Therefore, γ_{f}(s_{1}) ≠ γ_{f}(s_{2}). This implies that γ(s_{1}) ≠ γ(s_{2}), as required. □

Remark 2.2. In Theorem 2.1 we may choose X as having the same topological weight as S. That is, w(X) = w(S). Indeed, we can easily modify the second part of the proof, taking the family {X_{f} | f ∈ P }, where P separates the points of S.

§3. Eberlein groups and Ruppert’s problem.

Recall that for every semitopological semigroup S, the compactification u: S → S^{w} induced by the algebra W (S) is just the universal semitopological compactification of S.

Definition 3.1. We say that G is an Eberlein group if one of the following equivalent conditions holds:

(i) W (G) separates the points from the closed sets; (ii) u: G → G^{w} is a topological embedding; (iii) G is a topological subgroup of a compact semitopological semigroup.

For instance, every Hausdorff locally compact group is Eberlein.

Problem 3.2. (Ruppert [R1, p.114-115]) Find a Hausdorff topological group which is not Eberlein.

This problem is open even in the case when G is algebraically isomorphic to ℤ (cf. [R2]).

Every topological group G is a topological subgroup of Is(X)_{s} for a certain Banach space X (take, for example, X = C_{r}(G), the space for all right uniformly continuous functions on G (as in Teleman [T])). The natural question is: how good may X be ? When X may be Asplund or even reflexive? It turns out that the case of a reflexive X gives a characterization of Eberlein groups. Indeed, by combining Theorem 2.1 and Theorem 1.3 we obtain

Theorem 3.2. For every topological group G TFAE:

(i) G is an Eberlein group;

(ii) G is a topological subgroup of Is(X)_{s} for a certain reflexive Banach space X.

Below we discuss some related results and questions (having independent interest) which may help resolve Ruppert’s problem.

Proposition 3.3. Let G be a separable metrizable group and let U_{ℓ}(G) denote its left uniform structure. If G is Eberlein, then (G, U_{ℓ}(G)), as a uniform space, is embedded into a separable reflexive Banach space Y.

Proof. By Theorem 3.2, G is a topological subgroup of Is(X)_{s} for a certain reflexive Banach space X. Proceeding as in [Me2, Counterexample 3.13], without loss of generality we may suppose that X is separable. By the definition of the strong operator topology, the system of all orbit maps on G generates the uniformity U_{ℓ}(G). Since G is second countable, we may suppose that there exists a sequence z_{n} in X such that the corresponding sequence of orbit maps

z̃_{n}: G → X , z̃_{n}(g) = gz_{n}

generates U_{ℓ}(G). Moreover, we may suppose that ∥z_{n}∥ = 2^{−n}. Consider the ℓ_{2}- product ∏_{2} X_{n} = Y of the family {(X_{n}, ∥ ∥_{n}) | n ∈ ℕ}, where each (X_{n}, ∥ ∥_{n}) is a copy of (X, ∥ ∥). Clearly, Y is a reflexive Banach space. Since ∥z_{n}∥ = 2^{−n}, it is easy to show that the diagonal product map

γ: G → ∏_{2} X_{n} = Y, γ(g) = (gz_{n})

provides the desired uniform embedding. □

We say that a separable Banach space U is uniformly universal if every separable Banach space, as a uniform space, can be uniformly embedded into U. Aharoni [Ah] proved that c_{0} is uniformly universal. In response to a question by Yu. Smirnov, in 1969 Enflo [En] found a separable metrizable uniform space which is not uniformly embedded into a Hilbert space. That is, ℓ_{2} is not uniformly universal. However, it is not clear if here “Hilbert” may be replaced by “reflexive.”

Question 3.4. (a) Is it true that C[0, 1], c_{0} (or any other uniformly universal space) is a uniform subset of a certain reflexive Banach space?

(b) Equivalently, is it true that there exists a uniformly universal reflexive Banach space?

Note that if c_{0} is an Eberlein group, then, by Theorem 3.2 and Proposition 3.3, there exists a uniformly universal reflexive Banach space. Note also that there is no Lipschitz embedding of c_{0} into a reflexive Banach space [Ma]. For additional results in the theory of uniform (or Lipschitz) Banach embeddings, we refer to the survey of J. Lindenstrauss [Li].

In order to explain another possible link to Problem 3.2, let us recall that a Hausdorff topological group G is said to be minimal [Step] if there is no strictly coarser Hausdorff group topology on G.

Question 3.5. (Arhangel’skij (cf. [Di], [Me2]) Is it true that every Hausdorff topological group G is a quotient of a minimal topological group?

The following proposition implies a positive answer to Question 3.5 in the case of an Eberlein group.

Proposition 3.6. Every Eberlein group is a group retract of a minimal topological group.

Proof. In [Me1, Theorem 6.12] we proved that, if X is an Asplund space, then every topological subgroup G of Is(X)_{s} is a group retract of a minimal topological group. Now apply Theorem 3.2. □

If the answer to Arhangel’skij’s question or to Question 3.4 is negative, then this will provide the solution to Ruppert’s problem.

Acknowledgement. The author would like to thank A. Leiderman and V. Pestov for providing valuable information and for many stimulating discussions.

E-mail: megereli@macs.biu.ac.il

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