Topology Proceedings (To Appear)

REFLEXIVELY BUT NOT UNITARILY REPRESENTABLE TOPOLOGICAL GROUPS

MICHAEL G. MEGRELISHVILI

Abstract. We show that there exists a topological group G (namely, G := L_{4}[0, 1]) such that for a certain reflexive Banach space X the group G can be represented as a topological subgroup of Is(X) (the group of all linear isometries endowed with the strong operator topology) and such an X never may be Hilbert. This answers a question of V. Pestov and disproves a conjecture of A. Shtern.

1. Introduction

Let X be a real Banach space. Denote by Is(X)_{s} (Is(X)_{w}) the group of all linear isometries of X endowed with the strong (resp., weak) operator topologies.

A representation of a Hausdorff topological group G in X is a continuous group homomorphism G → Is(X)_{s}. Let K be a subclass of the class Ban of all Banach spaces. We say that G is K-representable if for a certain X ∈ K there exists a topological group embedding G ↪ Is(X)_{s}. Denote by K_{R} the class of all Krepresentable groups. For instance, this leads to the definitions of the following classes Ban_{R}, Ref_{R}, and Hilb_{R}, where Ref and Hilb denote all reflexive and all Hilbert spaces, respectively. We say that G is reflexively representable (unitarily representable) if G ∈ Ref_{R} (resp., G ∈ Hilb_{R}). Denote by TopGr the class of all Hausdorff topological groups. We have

TopGr = Ban_{R} ⊇ Ref_{R} ⊇ Hilb_{R}.

Indeed, the inclusions are trivial. As to TopGr = Ban_{R}, recall that every Hausdorff topological group G can be embedded into the group Is(E)_{s} of all linear isometries of a suitable Banach space E endowed with the strong operator topology. As in the paper of Teleman [23], take for example X := C_{r}^{b}(G), the Banach space of all bounded right uniformly continuous functions on G.

It is also well known that TopGr ≠ Hilb_{R}. Moreover, there are many examples of so-called exotic groups (that is, the groups whose unitary representations are trivial), see Herer-Christensen [12] and Banasczyk [3]. Among many interesting examples from [3], note that l_{2}/D is exotic for some discrete subgroup D of l_{2}. Not every separable Banach space (as a topological group) is unitarily representable. It is well known that l_{2} is not uniformly universal in the class of separable metrizable uniform spaces (Enflo [10], Aharoni [1]). These arguments lead to the fact that C[0, 1], c_{0} ∉ Hilb_{R} (see Proposition 3.7 below).

Recently, it has been proved in [18] that TopGr ≠ Ref_{R}. Denote by H_{+}[0, 1] the group of all orientation preserving selfhomeomorphisms of the closed interval endowed with the compact open topology. It turns out that every (weakly) continuous representation of H_{+}[0, 1] in a reflexive Banach space by linear isometries is trivial. This result answers a question discussed by Ruppert [20] and conjectured by V. Pestov.

2000 Mathematical Subject Classification. 22A25, 54H15, 43A65, 46B99.

A natural question arises about coincidence of Ref_{R} and Hilb_{R}. This question is posed in the recent paper by V. Pestov [19]. Earlier the positive answer was conjectured by A. Shtern [22]. The main result of the present work disproves this conjecture. Theorem 3.1 below implies that L_{4}[0, 1] ∈ Ref_{R} and L_{4}[0, 1] ∉ Hilb_{R}.

2. Weakly almost periodic functions

Recall that a continuous bounded function f ∈ C^{b}(G) on a topological group G is called weakly almost periodic (in short: wap) in the sense of Eberlein [9, 8] if the orbit of f in C^{b}(G) is relatively weakly compact. The subset W AP (G) of all wap functions in C^{b}(G) forms a closed (left and right) translation-invariant subalgebra. Every positive definite function is wap [5].

A semigroup S is semitopological if the multiplication S × S → S is separately continuous. The compactification j : G → G^{w} induced by the algebra W AP (G) is the universal semitopological compactification of G.

For every reflexive Banach space E, the semigroup

Θ(E)_{w} := {s ∈ L(E, E) : ∥s∥ ≤ 1}

of all contractive linear operators forms a compact semitopological semigroup in the weak operator topology [8]. Hence, the same is true for its closed subsemigroups. Conversely, an arbitrary compact Hausdorff semitopological semigroup can be obtained in this way (see Fact 2.2). By a result of Lawson [7, Corollary 6.3] every subgroup of a compact Hausdorff semitopological semigroup is a topological group.

Fact 2.1. Let G be a Hausdorff topological group. Then the following conditions are equivalent:

(i) There exists a reflexive Banach space E such that G is embedded as a topological subgroup into Is(E)_{s} (equivalently, G is reflexively representable);

(ii) There exists a reflexive Banach space E such that G is embedded as a topological subgroup into Is(E)_{w};

(iii) The algebra W AP (G) separates points and closed subsets;

(iv) The canonical map j : G → G^{w} is a topological embedding;

(v) G is a topological subgroup of a Hausdorff compact semitopological semigroup.

The equivalence of (iii), (iv) and (v) is well known [20, 5].

The part (ii) ⟹ (iii) follows from the fact that for every reflexive Banach space E and a norm-bounded semigroup S of linear operators on E the generalized matrix coefficients

{m_{v,f} : S → ℝ}_{f∈E^{∗},v∈E} m_{v,f}(s) = f(sv)

all are wap.

The part (iii) ⟹ (ii), is a direct consequence of the following result of A. Shtern.

Fact 2.2. ([22, 15, 17]) The following conditions are equivalent:

(a) W AP (S) separates points and closed subsets;

(b) S can be embedded into Θ(E)_{w} for a certain reflexive Banach space E.

As to the equivalence (i) ⟺ (ii), note that by [16, 17], strong and weak operator topologies coincide on Is(X) for a wide class of Banach spaces with P CP (the point of continuity property) including the class of all reflexive Banach spaces.

3. Main Results

Let (Ω, B, μ) be a measure space. The corresponding standard space L_{p}(Ω, B, μ) (1 ≤ p < ∞) will be denoted simply by L_{p}(μ).

Theorem 3.1. (i) L_{2k}(μ) ∈ Ref_{R} for every natural k ∈ ℕ; (ii) L_{p}[0, 1] ∉ Hilb_{R} for every 2 < p < ∞.

We say that a function F : A × B → ℝ has Double Limit Property (in short: DLP) if for every pair of sequences {a_{n}}, {b_{m}} in A and B respectively,

lim_{n} lim_{m} F (a_{n}, b_{m}) = lim_{m} lim_{n} F (a_{n}, b_{m})

whenever both of the limits exist.

We need Grothendieck’s characterization of wap in terms of Double Limit Property.

Fact 3.2. ([5, 20]) A function f ∈ C^{b}(G) is wap iff the induced map F : G×G → ℝ defined by F (g, h) := f(gh) has DLP, that is, for every pair of sequences {g_{n}}, {h_{m}} in G,

lim_{n} lim_{m} f(g_{n}h_{m}) = lim_{m} lim_{n} f(g_{n}h_{m})

whenever both of the limits exist.

Lemma 3.3. Let a topological group G admit a left-invariant metric d with DLP. Then G is reflexively representable.

Proof. Define the norm ∥g∥ := d(e, g). Then ∥g_{n}h_{m}∥ = d(g_{n}^{−1}, h_{m}).

By Grothendieck’s characterization (Fact 3.2), the bounded function

ϕ_{e} : G → ℝ g ↦ \frac{1}{1 + ∥g∥}

is wap. Then its left or right translations are also wap. Therefore, for every fixed z ∈ G the function

ϕ_{z} : G → ℝ g ↦ \frac{1}{1 + ∥gz∥}

is a wap function. Since the norm generates the original topology on G, the family {ϕ_{z}}_{z∈G} of wap functions separates points and closed subsets of G. Hence, by Fact 2.1 we can conclude that G ∈ Ref_{R}.

Lemma 3.4. The norm in the Banach space L_{2k}(μ) (k ∈ ℕ) has DLP.

Proof. We have to show that for every pair {u_{n}(t)}, {v_{m}(t)} of sequences in L_{2k}(μ)

lim_{n} lim_{m} ∥u_{n} + v_{m}∥ = lim_{m} lim_{n} ∥u_{n} + v_{m}∥

whenever both of the limits exist. We can suppose that the sequences are normbounded. Computing the norm ∥u_{n} + v_{m}∥, we get

∥u_{n} + v_{m}∥ = (∫_{Ω}(u_{n}(t) + v_{m}(t))^{2k}dμ)^{\frac{1}{2k}}

= (∥u_{n}∥^{2k} + ∑_{i=1}^{2k−1} C_{2k}^{i} < u_{n}^{2k−i}, v_{m}^{i} > +∥v_{m}∥^{2k})^{\frac{1}{2k}}

where

< u_{n}^{2k−i}, v_{m}^{i} >= ∫_{Ω} u_{n}^{2k−i}(t)v_{m}^{i}(t)dμ

and

u_{n}^{2k−i} ∈ L_{\frac{2k}{2k−i}} (μ), v_{m}^{i} ∈ L_{\frac{2k}{i}} (μ) = L∗_{\frac{2k}{2k−i}} (μ).

Passing to subsequences if necessary, we may assume that there exist

lim_{n} ∥u_{n}∥, lim_{m} ∥v_{m}∥.

By reflexivity, every bounded subset of L_{p}(μ) (p > 1) is relatively weakly compact and hence, relatively sequentially compact (as it follows by the classical Eberlein-Šmulian theorem). Therefore, we can suppose in addition (again by passing to subsequences) that there exist weak limits

weak − lim_{n} u_{n}^{2k−i}, weak − lim_{m} v_{m}^{i}

for every i ∈ {1, 2, ⋯, 2k − 1}.

Now, in order to complete the proof of the lemma it remains to observe that for every reflexive Banach space X and bounded subsets A ⊂ X, B ⊂ X^{∗}, the canonical duality

A × B → R, < x, f >= f(x)

has DLP. This fact easily follows using once again the Eberlein-Šmulian theorem.

By Lemmas 3.4 and 3.3 we get L_{2k}(μ) ∈ Ref_{R} ∀k ∈ N .

Now, we prove the second part of Theorem 3.1.

We need the following fundamental fact.

Fact 3.5. (Aharoni-Maurey-Mityagin [2]) For 2 < p < ∞, an infinite-dimensional L_{p}(μ) space is not uniformly embedded into a Hilbert space.

Additional information about uniform embeddings into Hilbert spaces can be found in [4].

The following Lemma is inspired by [14, Counterexample 2.13].

Lemma 3.6. The uniform space (Is(l_{2})_{s}, L), where L denotes the left uniformity on Is(l_{2})_{s}, can be uniformly embedded into l_{2}.

Proof. Since Is(l_{2})_{s} is separable and metrizable, there exists a sequence {v_{n}} in l_{2} such that:

(a): ∥v_{n}∥ = \frac{1}{2^{n}}

(b): {ṽ_{n} : Is(l_{2}) → l_{2}, g ↦ gv_{n}}_{n} generates the left uniformity on Is(l_{2}). Denote by B_{\frac{1}{2^{n}}} the closed \frac{1}{2^{n}} -ball centered at the origin with its usual uniformity.

Then the uniform product ∏_{n} B_{\frac{1}{2^{n}}} is uniformly embedded in a natural way into the l_{2}−sum (∑_{n}(l_{2})_{n})_{l_{2}} of countably many copies of the Hilbert space l_{2}. Eventually we have

Is(l_{2})_{s} ↪^{unif} ∏_{n} B_{\frac{1}{2^{n}}} ↪^{unif} (∑_{n}(l_{2})_{n})_{l_{2}} ⟷ l_{2}

Now we prove the second part (ii) of Theorem 3.1. Assuming the contrary, suppose that L_{p}[0, 1] (2 < p < ∞) is unitarily representable in the (infinitedimensional) Hilbert space H. Then, using the separability of L_{p}[0, 1] and passing to an appropriate separable infinite-dimensional closed linear subspace E of H, we can suppose even that there exists a topological group embedding L_{p}[0, 1] ↪ Is(E)_{s}. Since Is(E)_{s} and Is(l_{2})_{s} are topologically isomorphic, it is clear that Fact 3.5 and Lemma 3.6 will lead to a contradiction.

Therefore, we have L_{p}[0, 1] ∉ Hilb_{R} (2 < p < ∞) .

Theorem 3.1 is proved.

Proposition 3.7. C[0, 1], c_{0} ∉ Hilb_{R}.

Proof. Follows from Lemma 3.6 because l_{2} is not uniformly universal space for separable Banach spaces (Enflo [10]) in contrast to c_{0} (Aharoni [1]) or C[0, 1] (Banach-Mazur).

4. QUESTIONS

Question 4.1. Is it true that every Banach space X, as a topological group, is reflexively representable ? Or, equivalently, does W AP (X) separate points and closed subsets ?

By Lemma 3.4, the answer is yes for L_{p}(μ) spaces, where p = 2k is an arbitrary even integer. Using results of Shoenberg [21], we easily can extend this result to the case of 1 ≤ p ≤ 2. Indeed, the function f(v) = e^{−∥x∥^{p}} is positive definite (and hence, wap) on L_{p}(μ) spaces for every 1 ≤ p ≤ 2. Moreover, Chaatit [6] proved that every stable Banach space in the sense of Krivine-Maurey [13], in particular, every separable L_{p}(μ) space (1 ≤ p < ∞), is reflexively representable.

By Lemma 3.3, a closely related question is: for which Banach spaces does the original norm (or its some renorming) satisfy DLP ?

It is easy to show that the original norm of the Banach space c_{0} does not satisfy DLP. Indeed, define u_{n} := e_{n} (the standard basis vectors) and v_{m} := Σ_{i=1}^{m}e_{i}. Then the corresponding double limits are 1 and 2.

Lemma 3.3 suggests also the following questions:

Question 4.2. Let G be a reflexively representable group. Is it true that the topology of G is generated by a family of left-invariant pseudometrics with DLP ?

Question 4.3. Which (non-locally compact) metrizable topological groups admit a left-invariant metric with DLP ?

For instance, it would be interesting to know for which metric spaces (X, d) does the corresponding Graev’s metric ([11, 24]) on the free group F (X) satisfy DLP ?

Lemma 3.6 leads to the following natural question.

Question 4.4. Suppose a topological group G is such that the left uniformity of G admits a uniform embedding into l_{2}. Is then G unitarily representable ?

Acknowledgment. We thank to S. Gao, G. Godefroy and Y. Tseitlin for helpful suggestions. We are grateful to the referee for several improvements.

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Internet address: http://www.cs.biu.ac.il/˜ megereli

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

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