Group representations and construction of minimal topological groups

Michael Megrelishvili (Levy)* Department of Mathematics and Computer Science Bar-Ilan University 52 900 Ramat-Gan ISRAEL

Abstract. For every continuous biadditive mapping ω we construct a topological group M(ω) and establish its minimality under natural restrictions. Using the evaluation mapping G×G^{∗} → 𝕋 of Pontryagin-van Kampen duality and the canonical duality E×E^{∗} → ℝ for a normed space E, we obtain some new results in the theory of minimal groups. In particular, it is shown that every locally compact Abelian group is a group retract of a minimal locally compact group. Every Abelian topological group is a quotient of a perfectly minimal group.

Introduction.

A Hausdorff topological group G is said to be minimal (Stephenson [26]) if it does not admit a strictly coarser Hausdorff group topology. If G/P is minimal for every closed normal subgroup P of G then G is called totally minimal. A group G is perfectly minimal [12, §6.1] if the product G × X is minimal whenever X is minimal. Clearly, every compact group is totally minimal. At the same time, compact groups are perfectly minimal because every sup-complete (complete with respect to its two-sided uniformity) minimal group is perfectly minimal [14, Th. 3].

Recall some interesting examples of minimal groups:

a) The symmetric topological group S(X) of all bijections X → X of an infinite set X endowed with the topology τ_{X} of pointwise convergence is totally minimal (Dierolf and Schwanengel [10]). Moreover, every Hausdorff group topology on S(X) contains τ_{X} (Gaughan [17]).

Typeset by AMS-TEX

*Partially supported by the Israel Ministry of Sciences, Grant No. 3890.

1991 Mathematics Subject Classification. 22A05, 22B05.

Key words and phrases. Minimal group, Pontyagin-van Kampen duality, bilinear form, unconditionally closed subset.

b) Discrete minimal groups (i.e., groups which do not admit non-discrete Hausdorff group topologies). Such groups were constructed by Shelah [25] (assuming CH) and also by Hesse [18] and Ol’shanskii [22].

c) ℤ with the p-adic topology τ_{p} is a totally minimal non-perfectly minimal group (Doitchinov [13]).

d) The semidirect procuct ℝ λ_{π} ℝ_{+} of the multiplicative group ℝ_{+} of all positive numbers with ℝ under the natural action π : ℝ_{+} × ℝ → ℝ (Dierolf and Schwanengel [11]). This was generalized in Remus and Stoyanov [23]; in particular, it is proved that arbitrary powers of ℝ λ_{π} ℝ_{+} are minimal.

e) Every connected semi-simple Lie group is totally minimal iff the center is finite[23].

f) For every (real or complex) Hilbert space ℍ the unitary group U(ℍ) (which in this paper is always endowed with the strong operator topology) is totally minimal (Stoyanov [27]).

Prodanov and Stoyanov established that every Abelian minimal group is precompact (see [12,§2.7]). A general categorical approach to minimal topologies on algebraic structures was developed by Banaschewski [5]. For comprehensive information and terminology on minimal groups, we refer to [12].

In section 1 of this paper we introduce the notion of minimality for a biadditive mapping ω : E × F → A and prove that the canonical mapping Δ : G × G^{∗} → 𝕋 is minimal for every Hausdorff locally compact Abelian (LCA) group G. In this section we also establish that a minimal group G is perfectly minimal iff the center Z(G) is perfectly minimal.

In section 2 we define the induced group M(ω) and establish its minimality for minimal ω and A. As a consequence, we get that every LCA group is a group retract of a minimal locally compact (LC) group.

Section 3 contains a modification of our construction for bilinear forms. By the help of the canonical bilinear form ⟨ , ⟩: c_{0} × ℓ_{1} → ℝ and results from functional analysis a perfectly minimal group of countable weight is constructed which is not topologically isomorphic to a subgroup of the unitary group U(ℍ) for some Hilbert space ℍ (compare (f)). This solves a problem of Stoyanov [27].

In section 4, introducing the class BR of birepresentable groups (Definition 4.7), we show that BR contains: LC groups, subgroups of GL(E) (where E is a normed space and GL(E) is the group of all linear topological automorphisms endowed with the uniform operator topology), additive subgroups of locally convex vector spaces and free Abelian topological groups. The last result enables us to represent every Abelian topological group as a quotient of a perfectly minimal group. This gives a positive answer (in the realm of Abelian groups) to the following question of Arkhangel’skii ([4, Problem VI.6]): Is every Hausdorff group a quotient of a minimal group?

Using the same technique, we prove that every closed subgroup G of GL(n, ℝ) is a group retract of a minimal Lie group of dimension dim G + 2n + 1.

Some of our results were announced in [20, 21].

§1. Preliminaries.

Let (G, σ) and (X, τ) be topological groups. As usual, a representation α : G → Aut X is called continuous if the corresponding action α̃ : G×X → X, α̃(g, x) = α(g)(x) is continuous. In this situation, we say that X is a topological G-group. It is well-known that the semidirect product Xλ_{α}G is a topological group with respect to the product topology τ ×σ iff α is continuous. In the sequel we will often write e (the neutral element) instead of e_{G}, e_{X} or (e_{X}, e_{G}). We identify G with the subgroup {e} × G, as well as X with the normal subgroup X × {e}. Clearly, the canonical projection pr : Xλ_{α}G → G is a group retraction and ker pr = X. For further details on semidirect products see [24, Ch. 6].

The filter of all neighborhoods of an element x of a topological group (G, σ) will be denoted by N_{x}(G, σ), the zero element of an Abelian group X by o_{X}. Sometimes we write ∏ X_{i} instead of ∏_{i∈I} X_{i}. If P is a subgroup of G then σ|_{P} will denote the relative topology on P induced by σ and σ/P will be the quotient topology on the left coset space G/P. The following result is standard.

Lemma 1.1. ( [11], [12, Lemma 7.2.3]). Let P be a subgroup of a group G and let σ_{1} and σ_{2} be (not necessarily Hausdorff) group topologies on G with σ_{1} ⊂ σ_{2}, σ_{1}|_{X} = σ_{2}|_{X} and σ_{1}/X = σ_{2}/X. Then σ_{1} = σ_{2}.

Let E, F, A be Abelian groups. As usual, a mapping ω : E × F → A is called biadditive if the induced mappings ω_{x} : F → A, ω_{f} : E → A for every x ∈ E and f ∈ F are homomorphisms. Recall that ω is called separated if for every pair (x_{0}, f_{0}) of non-zero elements there exists a pair (x, f) such that f(x_{0}) ≠ o_{A} and f_{0}(x) ≠ o_{A}, where f(x) = ω(x, f). Note that in [16, Ch. X] a biadditive mapping is called “bilinear function.”

Definition 1.2. Let (E, σ), (F, τ), A be Abelian Hausdorff groups. A continuous separated biadditive mapping ω : (E, σ) × (F, τ) → A will be called minimal if for every coarser pair (σ_{1}, τ_{1}) of Hausdorff group topologies σ_{1} ⊂ σ, τ_{1} ⊂ τ such that ω : (E, σ_{1}) × (F, τ_{1}) → A is continuous (in such cases, we will say that (σ_{1}, τ_{1}) is ω-compatible), it follows that σ_{1} = σ and τ_{1} = τ.

Remark 1.3. Since ω is separated and A is Hausdorff, every ω-compatible pair (σ_{1}, τ_{1}) automatically is a pair of Hausdorff topologies. The contents of Definition 1.2 will not be changed if we omit the word “separated” provided that σ_{1}, τ_{1}, are not necessarily Hausdorff.

Example 1.4. Let A be a Hausdorff topological ring with a unit. Then the multiplication defines a minimal biadditive mapping ω : (A, +) × (A, +) → (A, +), where (A, +) is the additive group of A.

The following Lemma is elementary.

Lemma 1.5. Let ω : E × F → A be a continuous biadditive mapping, and P be a (not necessarily closed) subgroup of E. Then the natural mapping ω/P : E/P × P^{⊥} → A is continuous provided that P^{⊥} = {f ∈ F : f(x) = o_{A} ∀x ∈ P }.

Lemma 1.6. Let ω_{i} : E_{i} × F_{i} → A_{i} be a minimal biadditive mapping for every i ∈ I. Then the natural biadditive mapping ∏ ω_{i} : ∏ E_{i} × ∏ F_{i} → ∏ A_{i}, defined coordinatewise, is minimal. Proof. Denote by σ_{i} and τ_{i} the given topologies of E_{i} and F_{i} respectively. Let σ′ ⊂ σ = ∏ σ_{i} and τ′ ⊂ τ = ∏ τ_{i} be new coarser group topologies on ∏ E_{i} and ∏ F_{i} such that (σ′, τ′) is ∏ ω_{i}-compatible. In order to check the equalities σ′ = σ, τ′ = τ, it suffices to show that for every k ∈ I the projections

p_{k} : (∏ E_{i}, σ′) → (E_{k}, σ_{k}), q_{k} : (∏ F_{i}, τ′) → (F_{k}, τ_{k})

are continuous. Clearly, ker p_{k} = ∏{E_{i} : i ≠ k}. Set P_{k} = ker p_{k}. Since every ω_{i} is separated (Definition 1.2), we have P_{k}^{⊥} = F_{k}. By Lemma 1.5 (with ω := ∏ ω_{i}) we obtain that ω_{k} : (E_{k}, σ′/P_{k}) × (F_{k}, τ_{k}) → A_{k} is continuous. The minimality of ω_{k} implies σ′/P_{k} = σ_{k}. So every projection p_{k} is σ′-continuous. This means that σ′ = σ.

Analogously, using the dual version of Lemma 1.5, we obtain τ′ = τ. □ Lemma 1.7. Let (E, σ), (F, τ), A be Abelian Hausdorff groups and let ω : (E, σ) × (F, τ) → A be a continuous biadditive mapping. Denote by \overline{ω} : (\overline{E}, \overline{σ}) × (\overline{F} , \overline{τ}) → \overline{A} the corresponding completion [8, III, §6.5]. Then \overline{ω} is minimal if and only if ω is minimal. Proof. Sufficiency: Let ω be minimal and (ν, μ) be a \overline{ω}-compatible pair. Since (ν, μ) is coarser than (\overline{σ}, \overline{τ}), the minimality of ω implies ν|_{E} = \overline{σ}|_{E} = σ, μ|_{F} = \overline{τ}|_{F} = τ. Clearly, E is ν-dense and \overline{σ}-dense in \overline{E}. Therefore, ν/E and \overline{σ}/E are trivial topologies; in particular ν/E = \overline{σ}/E. Using Lemma 1.1 we get ν = \overline{σ}. Similarly, it can be proved that μ = \overline{τ}.

Necessity: If \overline{ω} is minimal, then according to Definition 1.2, \overline{ω} is separated. Since E and F are dense subgroups, then ω is separated too. Let (σ_{1}, τ_{1}) be a ω-compatible pair of Hausdorff group topologies. Consider the corresponding completion ω̂ : (Ê, σ̂_{1})× (F̂, τ̂_{1}) → \overline{A} and the natural mappings p : (\overline{E}, \overline{σ}) → (Ê, σ̂_{1}), q : (\overline{F} , \overline{τ}) → (F̂, τ̂_{1}), where p|_{E} = Id_{E} and q|_{F} = Id_{F} . Since E and F are dense and all groups are Hausdorff then the equality ω̂(p(x), q(f)) = \overline{ω}(x, f) holds for every (x, f) ∈ (\overline{E}, \overline{F}). Since \overline{ω} is separated, p and q are injective. Then (p^{−1}(σ̂_{1}), q^{−1}(τ̂_{1})) is a \overline{ω}-compatible pair of Hausdorff group topologies. The minimality of \overline{ω} implies p^{−1}(σ̂_{1}) = \overline{σ}, q^{−1}(τ̂_{1}) = \overline{τ}, which yields σ_{1} = σ, τ_{1} = τ. □ Definition 1.8. Let (G, τ) be a topological group and ψ = {ψ_{i} : G → Aut X_{i}}_{i∈I} be a system of representations in Hausdorff groups X_{i}. We say that ψ is topologically exact (or shortly: t-exact) if ψ is algebraically exact (i.e., ∩ ker ψ_{i} = {e}), τ is ψ-compatible (i.e., each ψ̃_{i} : (G, τ)×X_{i} → X_{i} is continuous) and every strictly coarser group topology τ′ is not ψ-compatible. ψ will be called hereditarily t-exact (ht-exact) if for every topological subgroup P of G the system {ψ_{i}|_{P} }_{i∈I} is t-exact. Remark 1.9.

(i) For a normed space (E, ∥ ∥) denote by Is(E) the group of all linear isometries of E onto itself endowed with the strong operator topology. Then the natural action Is(E) × E → E defines an ht-exact representation.

(ii) Let (G, τ) be an LC group and let Aut G be the group of all topological automorphisms of G endowed with the Birkhoff topology [19, §26]. Denote this topology by τ_{B} and recall that τ_{B} has a local base at the identity formed by the sets:

B(C, U) = {φ ∈ Aut G : φ(c) ∈ Uc and φ^{−1}(c) ∈ Uc ∀c ∈ C},

where C runs over the compact subsets of G and U runs over the neighborhoods of e in G. Then the natural action α : Aut G × G → G defines an ht-exact representation. Moreover, every α|_{P ×G}-compatible group topology σ on a subgroup P of Aut G is finer than τ_{B}|_{P} .

Indeed, fix a pair (C, U). Choose a neighborhood U_{1} ∈ N_{e}(G, τ) such that U_{1}U_{1}^{−1} ⊂ U. Since α|_{P ×G} : (P, σ) × (G, τ) → (G, τ) is continuous, for every c ∈ C there exist O(c) ∈ N_{c}(G, τ), V_{c} ∈ N_{e}(P, σ) such that V_{c}^{−1} = V_{c} and φ(x) ∈ U_{1}c for every x ∈ O(c) and φ ∈ V_{c}; in particular, O(c) ⊂ U_{1}c. Since C is compact there exists a finite subset

{c_{1}, c_{2}, …, c_{n}} of C such that C ⊂ ⋃_{i=1}^{n} O(c_{i}). Now, if we put V = ⋂_{i=1}^{n} V_{c_{i}}, then φ(c) ∈

U_{1}U_{1}^{−1}c ⊂ Uc for every c ∈ C and φ ∈ V = V^{−1}. Therefore, V ⊂ B(C, U) ∩ P. This means that τ_{B}|_{P} ⊂ σ.

Proposition 1.10. For every LCA group G the canonical biadditive mapping Δ : G^{∗} × G → 𝕋 is minimal. Proof. Let τ^{∗} and τ be the given (compact-open) topologies of G^{∗} and G and let (τ_{1}^{∗}, τ_{1}) be any Δ-compatible pair. For a τ-compact subset C of G and a number ε > 0 denote by [C, ε] the set {χ ∈ G^{∗} : χ(C) ⊂ O_{ε}}, where O_{ε} is the ε-neighborhood of the zero in 𝕋. Since C is τ_{1}-compact there exists U ∈ τ_{1}^{∗} such that e ∈ U and χ(C) ⊂ O_{ε} for every χ ∈ U. Therefore, e ∈ U ⊂ [C, ε], which yields τ_{1}^{∗} = τ^{∗}.

Using duality, we get τ_{1} = τ. □

Recall that a subset S of a group G is called unconditionally closed (u.c.) if S is closed for every Hausdorff group topology on G. The centralizer cen(g) of any element g ∈ G and the center Z(G) of G are u.c. in G (see [24, Lemma 3.11]). Lemma 1.11. Let X and Y be groups. Then X × Z(Y) and Z(X) × Y are u.c. in X × Y. Proof. Observe that X × Z(Y) = ∩{cen(e, y) : y ∈ Y } and Z(X) × Y = ∩{cen(x, e) : x ∈ X}. □

Now we need the following known result.

Proposition 1.12. [12, Proposition 7.2.5] Let G be a minimal group, and let H be a closed subgroup of the center Z(G) of G. Then H is minimal.

Proposition 1.13. Let X, Y be minimal groups. Then X × Y is minimal if and only if Z(X) × Z(Y) is minimal. Proof. Necessity: If X × Y is minimal then, by Proposition 1.12, Z(X × Y) (which coincides with Z(X) × Z(Y)) is minimal.

Sufficiency: Suppose that Z(X) × Z(Y) is minimal and γ_{1} ⊂ γ be a coarser Hausdorff group topology on X × Y. By Lemma 1.11 the subgroup Z(X) × Z(Y) is u.c. in X × Z(Y). The minimality of Z(X) × Z(Y) implies that Z(Y) is γ_{1}-closed in Z(X) × Z(Y). Therefore, Z(Y) is γ_{1}-closed in X × Z(Y). Since X is minimal, using Lemma 1.1 for the groups X × Z(Y) ⊃ Z(Y), we get γ_{1}|_{X×Z(Y)} = γ|_{X×Z(Y)}. In particular, X is γ_{1}-closed in X × Z(Y). On the other hand, X × Z(Y) is u.c. in X × Y (Lemma 1.11). Hence, X is γ_{1}-closed in X × Y. Using the minimality of Y and Lemma 1.1, we conclude γ_{1} = γ. □

Theorem 1.14. A minimal topological group X is perfectly minimal iff Z(X) is perfectly minimal. Proof. Suppose that Z(X) is perfectly minimal. For every minimal Y the center Z(Y) is minimal (Proposition 1.12). Therefore, Z(X) × Z(Y) is minimal. From Proposition 1.13 it follows that X × Y is minimal. So, X is perfectly minimal.

Conversely, let X be perfectly minimal. Then X × (ℤ, τ_{p}) is minimal for every prime p (see Introduction (c)). By Proposition 1.12 the groups Z(X) × (ℤ, τ_{p}) and Z(X) are minimal. Hence, Z(X) is precompact by the Prodanov-Stoyanov result (cf. Introduction). Now, Theorem 6.1.8 from [12] shows that Z(X) is perfectly minimal. □

Theorem 1.15. The arbitrary product (∏ X_{i}, σ = ∏ σ_{i}) of minimal groups (X_{i}, σ_{i}) with trivial center is perfectly minimal. Proof. By Theorem 1.14 it is sufficient to show that X = ∏ X_{i} is minimal. Since Z(X_{k}) = {e} then the kernel ker q_{k} = ∏{X_{i} : i ≠ k} of any projection q_{k} is u.c. in X (Lemma 1.11). Now, using the minimality of X_{k} we obtain that q_{k} is (ν, σ_{k})- continuous for every Hausdorff coarser group topology ν ⊂ σ. Thus, ν = σ. □

The last result gives an analog of Theorem 7.3.9(c) from [12] for minimal groups and, consequently, partially solves a problem posed in [12, p. 235]. §2. Induced groups of biadditive mappings.

Let E, F and A be Abelian topological groups and ω : E × F → A be a biadditive mapping. For all (f, a, x) ∈ F × A × E let ω^{▽}(f, (a, x)) = (a + f(x), x), where f(x) = ω(x, f). Since ω is biadditive, ω^{▽} defines an action of F on the product A × E. Moreover, every transition under this action is an automorphism. Denote the corresponding semidirect product A × E λ_{ω^{▽}}F by M(ω). We call this group the induced group of ω. Lemma 2.1.

(i) If u = (a, x, f), v = (b, y, φ) ∈ M(ω), then:

(a) uv = (a + b + f(y), x + y, f + φ);

(b) u^{−1} = (f(x) − a, −x, −f);

(c) uvu^{−1} = (f(y) − φ(x) + b, y, φ); (ii) If ω is separated then Z(M(ω)) = A. Proof (ii). Let u = (a, x, f) ∈ Z(M(ω)) and v = (o_{A}, o_{E}, φ). Then the equality uvu^{−1} = v implies φ(x) = o_{A} for every φ ∈ F. Since ω is separated, we conclude that x = o_{E}. Similarly, considering v = (o_{A}, y, o_{F}), we get f = o_{F} . On the other hand, clearly (a, o_{E}, o_{F}) ∈ Z(M(ω)) for every a ∈ A. □

Lemma 2.2.

(i) The action ω^{▽} : F × (A × E) → A × E is continuous iff ω is continuous. (ii) If ω is minimal then the induced representation defined by ω^{▽} is t-exact. Proof.

(i) It easily follows from the identity ω^{▽}(f, (a, x)) − (a, x) = (f(x), o_{E}). (ii) Let τ_{1} ⊂ τ be a ω^{▽}-compatible group topology on F. By (i) the mapping ω : E × (F, τ_{1}) → A is continuous. The minimality of ω implies τ_{1} = τ. □

Proposition 2.3. For every LCA group G the mappings

i_{1} : G → Aut(𝕋 × G^{∗}) , i_{1}(g)(t, χ) = (t + χ(g), χ),
i_{2} : G^{∗} → Aut(𝕋 × G) , i_{2}(χ)(t, g) = (t + χ(g), g)

are topological group embeddings.

Proof. By Proposition 1.10, Δ : G^{∗} × G → 𝕋 is minimal. Combining Lemma 2.2(ii) and Remark 1.9(ii) we obtain that i_{1} is a topological embedding. Using duality we get our assertion for i_{2}. □ Definition 2.4. Let q : X → Y be a (not necessarily group) retraction of a group X on a subgroup Y. We say that q is central if q(xyx^{−1}) = y for each (x, y) ∈ X × Y.

Lemma 2.5. For every biadditive mapping ω : E × F → A the natural projections q_{E} : M(ω) → E, q_{F} : M(ω) → F are central. Proof. If u = (a, x, f) ∈ M(ω), y ∈ E and φ ∈ F then by Lemma 2.1(c) holds uyu^{−1} = (f(y), y, o_{F}), uφu^{−1} = (−φ(x), o_{E}, φ). □

Proposition 2.6. Let (M, γ) be a topological group such that M is algebraically a semidirect product M = Xλ_{α}G. If q : X → Y is a continuous central retraction of the topological subgroup X on a topological G-subgroup Y of X, then the action

α|_{G×Y} : (G, γ/X) × (Y, γ |_{Y}) → (Y, γ |_{Y})  (1)

is continuous. Proof. Clearly, each g-transition (Y, γ |_{Y}) → (Y, γ |_{Y}) is continuous. We have to show that α|_{G×Y} in (1) is continuous at (e, y) for every y ∈ Y. Fix an arbitrary y ∈ Y and a neighborhood O(y) ∈ N_{y}(Y, γ |_{Y}). Since q : (X, γ |_{X}) → (Y, γ |_{Y}) is continuous, there exists U_{1} ∈ N_{e}(M, γ) such that

q(U_{1} ∩ X) ⊂ O.  (2)

Choose V, U_{2} ∈ N_{e}(M, γ) with the property

vU_{2}v^{−1} ⊂ U_{1} ∀v ∈ V.  (3)

Now, if pr : M → G denotes the canonical projection, then

α(g, z) ∈ O ∀g ∈ pr(V), ∀z ∈ U_{2} ∩ Y,

Indeed, if v = (x, g) ∈ V and z ∈ U_{2} ∩ Y then by (3), vzv^{−1} ∈ U_{1}. ¿From the normality of X we have vzv^{−1} ∈ X. Thus, vzv^{−1} ∈ U_{1} ∩ X. Using (2) and the identity vzv^{−1} = xα(g, z)x^{−1}, we get q(xα(g, z)x^{−1}) ∈ O. Since q is central and α(g, z) ∈ Y, we obtain α(g, z) ∈ O. Finally, observe that pr(V) ∈ N_{e}(G, γ/X) and U_{2} ∩ Y ∈ N_{y}(Y, γ |_{Y}). □

Proposition 2.7. Let (Xλ_{α}G, γ) = M be a topological semidirect product and {Y_{i}}_{i∈I} be a system of G-subgroups in X such that the system {α|_{G×Y_{i}} : G × Y_{i} → Y_{i}}_{i∈I} is texact and for each i ∈ I there exists a continuous central retraction q_{i}: X → Y_{i}. Suppose that γ_{1} ⊂ γ is a coarser group topology on M such that γ_{1} |_{X} = γ |_{X} . Then γ_{1} = γ. Proof. Proposition 2.6 shows that each action α|_{G×Y_{i}} : (G, γ_{1}/X)×Y_{i} → Y_{i} (i ∈ I) is continuous. By Definition 1.8, γ_{1}/X coincides with the given topology of G. Therefore, γ_{1}/X = γ/X. Now, Lemma 1.1 completes the proof. □

Corollary 2.8. Let (Xλ_{α}G, γ) be a topological semidirect product and let α be t-exact. Suppose that X is Abelian and γ_{1} ⊂ γ is a group topology which agrees with γ on X. Then γ_{1} = γ. Proof. Since X is Abelian, the identity mapping Id_{X} : X → X is central. □

The commutativity of X is essential. It was shown in ([14, Example 10]) that there exists a totally minimal precompact torsion group X such that a certain semidirect product XλG is not minimal, where G is the discrete cyclic group of order 2. Proposition 2.9. Let ω : (E, σ) × (F, τ) → A be a minimal biadditive mapping and (M(ω), γ) = (A × E λ_{ω^{▽}}F, γ) be the induced group. If γ_{1} ⊂ γ is a group topology such that γ_{1} |_{A} = γ |_{A} , then γ_{1} = γ. Proof. By Lemma 1.1 it suffices to show that γ_{1}/A = γ/A. Therefore, we have to check the continuity of the projection

q_{E×F} : (M(ω), γ_{1}) → (E × F, σ × τ).  (4)

By Proposition 2.6 (with X := A × E, q := Id_{X}, G := F) the action ω^{▽} : (F, γ_{1}/A × E) × (A × E, γ_{1} |_{A×E}) → (A × E, γ_{1} |_{A×E}) is continuous. Therefore, by Lemma 2.2 (i) the biadditive mapping

ω : (E, γ_{1} |_{E}) × (F, γ_{1}/A × E) → (A, γ_{1} |_{A})

is continuous. From the equality γ_{1} |_{A} = γ |_{A} and the minimality of ω, it follows that the topology γ_{1}/A × E on F coincides with the given topology τ. Thus, the projection q_{F} : (M(ω), γ_{1}) → (F, τ) is continuous. Now, observe that A × F (which is identified with A × {e} × F) is a normal subgroup of M and (A × F) ∩ E = {e}. This means (see [24, Definition 6.10]) that M(ω) is algebraically a semidirect product A × F λ E. Moreover, the corresponding action (by means of the inner automorphisms) of E on A × F is

ω_{∗}^{▽} : E × A × F → A × F, ω_{∗}^{▽}(x, (a, f)) = (a + ω_{∗}(f, x), f),

where ω_{∗} : F × E → A is a biadditive mapping defined by w_{∗}(f, x) = −ω(x, f). Clearly, ω_{∗} is minimal iff ω is minimal. Therefore, as in the case of q_{F} , using Proposition 2.6 (with X := A × F, q := Id_{X}, G := E) and Lemma 2.2(i) (for ω_{∗}), we can establish the continuity of q_{E} : (M(ω), γ_{1}) → (E, σ). Thus, q_{E×F} from (4) is continuous. □

Theorem 2.10. Let ω : E × F → A be a minimal biadditive mapping. Then the induced group M(ω) is (perfectly) minimal iff A is (perfectly) minimal. Proof. The “minimal case” follows from Proposition 1.12, Proposition 2.9 and Lemma 2.1(ii), for the “perfectly minimal case” we use, in addition, Theorem 1.14. □

In the sequel, instead of M(Δ) we will write M(G). Theorem 2.11. Let Δ : G^{∗} × G → 𝕋 be the canonical biadditive mapping for an LCA group G. Then the induced group M(G) = 𝕋 × G^{∗}λ G is minimal. In particular, every LCA group is a group retract of a minimal LC group. Proof. Apply Proposition 1.10 and Theorem 2.10. □

Remus and Stoyanov [23] proved that every compactly generated LCA group is a quotient of a minimal LC group.

Theorem 2.12. Every locally precompact Abelian group G is a group retract of a locally precompact perfectly minimal group. Proof. Consider the natural mapping Δ : (\overline{G})^{∗} × \overline{G} → 𝕋, where \overline{G} is the completion of G. By Proposition 1.10 and Lemma 1.7 the restricted mapping Δ|_{(\overline{G})^{∗}×G} : (\overline{G})^{∗}×G → 𝕋 is minimal. Now, it follows from Theorem 2.10 that the induced group 𝕋 × (\overline{G})^{∗}λ G is perfectly minimal. □ Question 2.13.

(i) Let \underline{A} be a certain class of topological groups and \underline{min} denotes the class of all minimal groups. The general questions which naturally arise here are the following: Is it true that every G ∈ \underline{A} is a group retract of a group M ∈ \underline{A}∩\underline{min}? What happens if \underline{A} is the class of all locally compact (or Lie) groups?

(ii) Arkhangel’skii [4, Problem VI.6] posed the question if every Hausdorff group is a quotient of a minimal group. V. Uspenskii announced a positive answer (see [29], Theorem 3.3E.2).

In the sequel we give some partial results concerning Question 2.13. The following is one of them. Theorem 2.14. An arbitrary product of LCA groups is a group retract of a minimal group M which may be represented as a product of LC groups.

The proof follows from Proposition 1.10 and the following generalization of Theorem 2.10. Theorem 2.15. Let for every i ∈ I, ω_{i} : E_{i} × F_{i} → A_{i} be a minimal biadditive mapping. Then ∏ M(ω_{i}) is (perfectly) minimal iff ∏ A_{i} is (perfectly) minimal. Proof. The natural mapping

∏ M(ω_{i}) → M (∏ ω_{i}) , (a_{i}, x_{i}, f_{i})_{i∈I} → ((a_{i})_{i∈I}, (x_{i})_{i∈I}, (f_{i})_{i∈I})

is a topological isomorphism. Now, our assertion follows from Lemma 1.6 and Theorem 2.10 (for ∏ ω_{i}). □

§3. Bilinear forms and induced groups.

As usual, the dual space (E^{∗}, ∥ ∥^{∗}) of a real normed space (E, ∥ ∥) is the Banach space of all continuous linear functionals f : E → ℝ equipped with the norm ∥ f ∥^{∗}= sup{|f(x)|: ∥ x ∥≤ 1}.

Definition 3.1. Let E, F be normed spaces. We say that a continuous bilinear form ω : E × F → ℝ is a strong duality if for every norm-unbounded sequences (x_{n}) ⊂ E and (f_{n}) ⊂ F the subsets S_{1} = {f(x_{n}) :∥ f ∥≤ 1, n ∈ ℕ}, S_{2} = {f_{n}(x) : n ∈ ℕ, ∥ x ∥≤ 1} are unbounded in ℝ.

Remark 3.2. Considering in Definition 3.1 the sequences x_{n} = nx_{0} and f_{n} = nf_{0}, where (x_{0}, f_{0}) ∈ E × F, we obtain that every strong duality is a separated biadditive mapping.

Lemma 3.3. For every normed space E the canonical bilinear form ⟨ , ⟩ : E ×E^{∗} → ℝ defined by ⟨x, f⟩ = f(x), is a strong duality.

Proof. Suppose (x_{n}) ⊂ E and (f_{n}) ⊂ E^{∗} are norm-unbounded. For every x_{n} there exists φ_{n} ∈ E^{∗} such that ∥ φ_{n} ∥^{∗}= 1 and φ_{n}(x_{n}) =∥ x_{n} ∥ (cf. [6, Theorem (40.10)]. Hence, S_{1} (from Definition 3.1) is unbounded. The unboundedness of S_{2} directly follows from the definition of ∥ ∥^{∗} . □

Proposition 3.4. Let G be an LC group and μ be the standard Haar measure on G. Let (L^{1}(G), ∥ ∥_{1}) be the Banach space of all equivalence classes of 1-integrable real functions on G, and denote by (K(G), ∥ ∥) the normed space of all continuous real functions with compact supports endowed with the sup-norm. Then the natural bilinear form ω: L^{1}(G) × K(G) → ℝ, ω(f, φ) = ∫_{G} fφdμ is a strong duality.

Proof. Considering for every φ_{0} ∈ K(G) the corresponding functional φ̂_{0} : L^{1}(G) → ℝ, φ̂_{o}(f) = ∫_{G} fφ_{0}dμ we obtain an embedding of K(G) into the normed space (L^{1}(G))^{∗} = L^{∞}(G) (see the remark in [8, IV, §6, 3]). Therefore, the norm ∥ φ_{o} ∥ is equal to the number

∥ φ_{0} ∥= sup{ω(f, φ_{0}): f ∈ L^{1}(G), ∥ f ∥_{1}≤ 1}.  (1)

On the other hand, by [19, Theorem (12.13)], we have

∥ f_{o} ∥_{1}= sup{ω(f_{o}, φ): φ ∈ K(G), ∥ φ ∥≤ 1}.  (2)

Clearly, (1) and (2) imply that ω is a strong duality. □

Lemma 3.5. Let (E, ∥ ∥) be a normed space and let X ⊂ E be an additive topological subgroup. Suppose that τ is a strictly coarser group topology on X. Then every τneighborhood U ∈ N_{0}(X, τ) of the zero in X is norm-unbounded. Proof. Since τ is strictly coarser, there exists ε_{0} > 0 such that every O ∈ N_{0}(X, τ) contains an element x ∈ O with ∥ x ∥≥ ε_{0}. Fix U ∈ N_{0}(X, τ). Since τ is a group topology, for each natural n there exists V_{n} ∈ N_{0}(X, τ) such that nV_{n} ⊂ U. Choose x_{n} ∈ V_{n} with the property ∥ x_{n} ∥≥ ε_{0}. Then ∥ nx_{n} ∥≥ nε_{0}. Since nx_{n} ∈ U, this means that U is norm-unbounded. □

Proposition 3.6. Every strong duality is a minimal biadditive mapping. Proof. Apply Lemma 3.5. □

Let E, F be normed spaces and ω : E × F → ℝ be a continuous bilinear form. Consider the induced group M(ω) = A × E λ_{ω^{▽}}F of the biadditive mapping ω and define the following action:

π : ℝ_{+} × M(ω) → M(ω), π(t, (a, x, f)) = (ta, tx, f).

Clearly, π defines a continuous representation. Therefore, the topological semidirect product M(ω) λ_{π} ℝ_{+} is well-defined. Denote this group by M_{+}(ω) and call it the induced group of the bilinear form ω. In the case of the canonical bilinear form ⟨ , ⟩ : E ×E^{∗} → ℝ we will use the notation M_{+}(E) instead of M_{+}(⟨ , ⟩).

The proof of the following result is straightforward. Proposition 3.7. Let E be a normed space. Then the following mappings:

i_{1} : E^{∗} × ℝ_{+} → GL(ℝ × E) , i_{1}(f, t)(a, x) = (ta + tf(x), tx),
i_{2} : E × ℝ_{+} → GL(ℝ × E^{∗}) , i_{2}(x, t)(a, f) = (ta + tf(x), tf)

are topological group embeddings.

Lemma 3.8.

(i) If u = (a, x, f, t), v = (b, y, φ, s) ∈ M_{+}(ω) then uvu^{−1} = (a − sa + tf(y) − sφ(x) + tb, x − sx + ty, φ, s),

(ii) Z(M_{+}(ω)) = {e}, for every separated bilinear form ω.

Proof (ii). By (i), it is clear that Z(M_{+}(ω)) ⊂ Z(M(ω)). According to Lemma 2.1(ii), Z(M(ω)) = ℝ. Therefore, every element from Z(M_{+}(ω)) has the form v = (b, o_{E}, o_{F} , 1). The equality uvu^{−1} = v for u = (o, o_{E}, o_{F} , 2) implies 2b = b. Thus, b = o. □

Theorem 3.9. Let E, F be normed spaces and let ω : E × F → ℝ be a bilinear form. If ω is minimal as a biadditive mapping (in particular if ω is a strong duality), then the induced group M_{+}(ω) is perfectly minimal.

Proof. By Theorem 1.14 and Lemma 3.8(ii) it suffices to verify the minimality of M_{+}(ω). Denote by γ the given topology of M_{+}(ω) and suppose that γ_{1} ⊂ γ is a Hausdorff group topology. The minimal group ℝ λ ℝ_{+} (Introduction (d)) is naturally embedded in M_{+}(ω). Therefore, γ_{1} |_{ℝ} = γ |_{ℝ} . From Proposition 2.9 immediately follows γ_{1} |_{M(ω)} = γ |_{M(ω)} . Now observe that the natural retraction q : M(ω) → ℝ is central (moreover, by Lemma 2.1(ii), ℝ = Z(M(ω))) and the action of ℝ_{+} on ℝ is t-exact. By Proposition 2.7 (in the situation: G := ℝ_{+}, X := M(ω), Y := ℝ) we get γ_{1} = γ. □

Theorem 3.10. For every normed space E the induced group M_{+}(E) = (ℝ×E λ E^{∗})λ_{π} ℝ_{+} of the canonical duality E × E^{∗} → ℝ is perfectly minimal.

Proof. This is immediate from Lemma 3.3 and Theorem 3.9. □

In the sequel the left uniformity of a group G is denoted by U_{ℓ}(G).

Corollary 3.11. Every (metrizable) uniform space X is uniformly embedded as a closed U_{ℓ}(M)-uniform subspace into a (metrizable) perfectly minimal group M.

Proof. Every metric space is isometric to a closed subspace of a normed space (Arens - Eells [3]). In the general case we consider a closed uniform embedding X → ∏ E_{i} into a product of normed spaces and apply Theorem 1.15. □

Recall that the dual space of the Banach space c_{0} is ℓ_{1}. Consider the induced group M(c_{0}) = (ℝ × c_{0} λ ℓ_{1})λ ℝ_{+}. By a result of Aharoni [1] every separable metrizable uniform space is uniformly embedded in c_{0}. Thus, we obtain the following:

Theorem 3.12. Every separable metrizable uniform space is uniformly embedded into the perfectly minimal separable metrizable group M(c_{0}).

Now we give a counterexample to a conjecture of Stoyanov ([27], [12, p. 263]).

Counterexample 3.13. The (perfectly) minimal group M(c_{0}) is not topologically isomorphic to a subgroup of the unitary group U(ℍ) of a Hilbert space ℍ.

Proof. In [15] Enflo constructs a countable metric space which is not uniformly embeddable into a Hilbert space. Hence, by Theorem 3.12, M(c_{0}) is not uniformly embedded in a Hilbert space. Therefore, it suffices to show that every topological subgroup G of U(ℍ), such that G has countable weight, is U_{ℓ}(G)-embedded in ℓ_{2}. Suppose that G is such a group. Since the left uniformity on U(ℍ) is induced by the strong operator topology (see [12, p. 246]), the system {α_{x} : G → ℍ}_{x∈ℍ} of all orbit mappings (α_{x}(g) = gx) induces U_{ℓ}(G). Since G has countable weight, there exists a countable subset {x_{n} : n ∈ ℕ} of ℍ such that {α_{x_{n}}}_{n∈ℕ} generates U_{ℓ}(G). From this fact it follows that (G, U_{ℓ}(G)) is uniformly embedded in the uniform space ℍ^{ℵ_{0}}. For every natural n the Hilbert subspace ⟨Gx_{n}⟩ generated by the orbit of x_{n} is separable. Thus, without loss of generality, we can assume that ℍ is separable. Hence, in order to complete our proof it suffices to establish that ℓ_{2}^{ℵ_{0}} is uniformly embedded in ℓ_{2}. By a result of Aharoni [2] ℓ_{2} is uniformly embedded into a bounded subset of itself. Hence, ℓ_{2}^{ℵ_{0}} is uniformly embedded into the product ∏{B_{2^{−n}} : n ∈ ℕ} of 2^{−n}-balls. By definition, this product is a uniform subspace of the Hilbert sum ⨁_{n∈ℕ}(ℓ_{2})_{n} being isomorphic to ℓ_{2}. □

§4. Group representations in biadditive mappings.

Let G be a topological group and let ω : E × F → A be a biadditive mapping. A continuous birepresentation of G (or: G-birepresentation) in ω is a pair (α_{1}, α_{2}) of continuous actions α_{1} : G × E → E and α_{2} : G × F → F such that ω is G-invariant (i.e., ω(gx, gf) = ω(x, f)).

In the case of normed spaces E and F, a bilinear form ω : E × F → ℝ and linear actions α_{1}, α_{2} we obtain the definition of a linear birepresentation.

We will say that a linear representation α : G × E → E in a normed space E is bicontinuous if the dual action α^{∗} : G × E^{∗} → E^{∗}, α^{∗}(g, φ)(x) = φ(g^{−1}x) is continuous. In other words, a linear action is bicontinuous iff the induced canonical birepresentation is continuous. Not every continuous representation is bicontinuous even in the case of norm-invariant actions of a compact group. Remark 4.1. Every continuous representation of a topological group G by unitary operators on a Hilbert space ℍ is bicontinuous. It is also clear that for every normed space E the natural representation GL(E) × E → E is bicontinuous.

Let ψ = {ψ_{i}}_{i∈I} be a system of continuous G-birepresentations:

ψ_{i} = (ω_{i} : E_{i} × F_{i} → A_{i}, α_{1i} : G × E_{i} → E_{i}, α_{2i} : G × F_{i} → F_{i}).

By the induced group M(ψ) of a system ψ we mean the topological semidirect product ∏ M(ω_{i}) λ_{π} G, where the action

π : G × ∏ M(ω_{i}) → ∏ M(ω_{i})

is defined coordinate-wise by means of the following system {π_{i}}_{i∈I} of actions:

π_{i} : G × M(ω_{i}) → M(ω_{i}), π_{i}(g, (a, x, f)) = (a, gx, gf),

where gx = α_{1i}(g, x), gf = α_{2i}(g, f).

Analogously, in the case of linear birepresentations, we define the induced group M_{+}(ψ) as the semidirect product ∏ M_{+}(ω_{i}) λ_{π} G considering

π_{i} : G × M_{+}(ω_{i}) → M_{+}(ω_{i}), π_{i}(g, (a, x, f, t)) = (a, gx, gf, t).

Definition 4.2. A system ψ = {ψ_{i}}_{i∈I} is called (ht-exact) t-exact if the system {α_{ki} : k ∈ {1, 2}, i ∈ I} of actions is (ht-exact) t-exact in the sense of Definition 1.8.

Theorem 4.3. Assume that ψ = {ψ_{i}}_{i∈I} is a t-exact system of G-birepresentations such that every ω_{i} : E_{i}×F_{i} → A_{i} is a minimal biadditive mapping and ∏ A_{i} is (perfectly) minimal. Then M(ψ) is (perfectly) minimal. Proof. By our construction, M(ψ) = ∏ M(ω_{i}) λ_{π} G. Let γ_{1} ⊂ γ be a coarser Hausdorff group topology, and let X denote the subgroup ∏ M(ω_{i}). The minimality of X (Theorem 2.15) implies γ_{1} |_{X} = γ |_{X} . By Lemma 2.5, the retractions M(ω_{i}) → E_{i}, M(ω_{i}) → F_{i} are central. Then the natural projections X → E_{i}, X → F_{i} are central, too. Using Proposition 2.7, we obtain γ_{1} = γ. This proves the minimality of M(ψ).

In the “perfectly minimal case,” due to Theorem 1.14, it remains to show that Z(M(ψ)) = ∏ A_{i}. Let u = ((a_{i})_{i∈I}, (x_{i})_{i∈I}, (f_{i})_{i∈I}, g) ∈ Z(M(ψ)) and k ∈ I. Now, if y ∈ E_{k} ⊂ M(ψ) and f ∈ F_{k} ⊂ M(ψ), then the equalities uy = yu, uf = fu and the elementary computations (see Lemma 2.1(i)) show that α_{1k}(g, y) = y, α_{2k}(g, f) = f. By exactness of ψ, we conclude that g = e. This means that Z(M(ψ)) ⊂ Z(X). It follows from Lemma 2.1(ii) that Z(X) = ∏ A_{i}. Therefore, Z(M(ψ)) ⊂ ∏ A_{i}. Since the action π|_{G×∏ A_{i}} is trivial, then Z(M(ψ)) = ∏ A_{i}. □

Theorem 4.4. Let (G, τ) be an LCA group and P be a topological subgroup of (Aut G, τ_{B}), where τ_{B} is the Birkhoff topology. Consider the action α: P × G → G, α(φ, g) = φ(g) and the dual action α^{∗}: P × G^{∗} → G^{∗}, α^{∗}(φ, χ)(g) = χ(φ^{−1}(g)). Denote by ψ the P - birepresentation in Δ: G^{∗} × G → 𝕋 defined by the pair (α^{∗}, α). Then the induced group M(ψ) = M(G) λ_{π} P is perfectly minimal. Proof. Follows from Remark 1.9(ii), Proposition 1.10 and Theorem 4.3. □

Corollary 4.5. Every closed subgroup G of GL(n, ℝ) is a group retract of a minimal Lie group M of dimension dim G + 2n + 1. Proof. Using the selfduality (ℝ^{n})^{∗} = ℝ^{n} we get from Theorem 4.4 that the group M = (𝕋 × ℝ^{n} λ_{α} ℝ^{n}) λ_{π} G is minimal. In order to show that M is a Lie group, it suffices to check the analyticity of α and π. Let ⟨ , ⟩: ℝ^{n} × ℝ^{n} → ℝ be the canonical scalar product and let q: ℝ → 𝕋 be the natural homomorphism. Then α(u, (t, v)) = (q(⟨u, v⟩) + t, v). This description shows that α is analytic. The action π is defined by the rule π(g, (t, v, u)) = (t, ν(g, v), ν(g, u)), where ν is a restriction of the action GL(n, ℝ) × ℝ^{n} → ℝ^{n} to the submanifold G × ℝ^{n}. Thus, π is analytic, too. □

Theorem 4.6. Let for every i ∈ I, E_{i}, F_{i} be normed spaces, ω_{i} : E_{i} × F_{i} → ℝ be strong dualities and ψ = {ψ_{i}}_{i∈I} be a t-exact system of linear G-birepresentations. Then M_{+}(ψ) is perfectly minimal.

Proof. Let X_{+} and X denote ∏ M_{+}(ω_{i}) and ∏ M(ω_{i}) respectively. Every Z(M_{+}(ω_{i})) is trivial (Lemma 3.8(ii)). Therefore, Z(X_{+}) = {e}. As in the proof of Theorem 4.3, elementary computations (using Lemma 3.8(i)) show that Z(M_{+}(ψ)) ⊂ Z(X_{+}). Thus, Z(M_{+}(ψ)) = {e}. By Theorem 1.14 it suffices to check the minimality of M_{+}(ψ). Let γ_{1} ⊂ γ be a coarser Hausdorff group topology. Since each M_{+}(ω_{i}) is minimal (Theorem 3.9), then Theorem 1.15 establishes the minimality of X_{+}. Therefore, γ_{1} |_{X_{+}} = γ |_{X_{+}} . In particular, γ_{1} |_{X} = γ |_{X} .

Now, as in the proof of Theorem 4.3, we use central retractions. It follows from Lemma 2.5 that the natural projections X → E_{i}, X → F_{i} are central. Using Proposition 2.7 we obtain that γ_{1} agrees with γ on X λ G. Arbitrary powers of the group ℝ λ ℝ_{+} are minimal (see Introduction (d) or Theorem 1.15). Hence, the “product” action of ∏(ℝ_{+})_{i} on ∏ ℝ_{i} is t-exact. Taking this fact into account, we use Proposition 2.7 in the following situation: G := ∏(ℝ_{+})_{i}, X := ∏ M(ω_{i}) λ G, Y := ∏ ℝ_{i} and q : X → Y is the natural projection. Hence we obtain γ_{1} = γ. □

Definition 4.7. We say that a topological group G is an (HBR-group) BR-group if there exists an (ht-exact) t-exact system {ψ_{i}}_{i∈I} of linear G-birepresentations, such that every ω_{i} : E_{i} × F_{i} → ℝ is a strong duality.

Theorem 4.8. For every BR-group G there exists a continuous group retraction p : M → G such that M and ker p are perfectly minimal. Proof. Take the perfectly minimal group M: = M_{+}(ψ) from Theorem 4.6 and consider the projection p: M_{+}(ψ) → G. Then ker p coincides with ∏ M_{+}(ψ_{i}) which is perfectly minimal by Theorem 3.9, Lemma 3.8(ii) and Theorem 1.15. □

Let ω: L^{1}(G) × K(G) → ℝ be the bilinear form from Proposition 3.4. The pair of actions:

α_{1} : G × L^{1}(G) → L^{1}(G),  α_{1}(g, f)(x) = f(g^{−1}x)
α_{2} : G × K(G) → K(G),  α_{2}(g, φ)(x) = φ(g^{−1}x)

defines a continuous birepresentation in ω. Moreover, since α_{2} induces a topological group embedding G → Is(K(G)), then this birepresentation is ht-exact. Thus, LC is a subclass of HBR. Lemma 4.9. Let {ω_{i}: E_{i} × F_{i} → ℝ}_{i∈I} be a system of strong dualities, {(G_{i}, τ_{i})}_{i∈I} be a system of topological groups, and for every i ∈ I let ψ_{i} be a continuous linear G_{i}-birepresentation ψ_{i} = (ω_{i}: E_{i} × F_{i} → ℝ, α_{1i}: G_{i} × E_{i} → E_{i}, α_{2i}: G_{i} × F_{i} → F_{i}). Suppose that for every i ∈ I the family {(α_{1i})_{x}: G_{i} → E_{i}, (α_{2i})_{f}: G_{i} → F_{i}}_{x∈E_{i}, f∈F_{i}} of all orbit mappings generates the topology τ_{i}. Then the product (G, τ) = ∏(G_{i}, τ_{i}) is an HBR-group. Proof. For every i ∈ I define the actions:

α_{1i}^{Π} : G × E_{i} → E_{i} , α_{1i}^{Π} ((g_{s})_{s∈I}, x_{i}) = α_{1i}(g_{i}, x_{i})
α_{2i}^{Π} : G × F_{i} → F_{i} , α_{2i}^{Π} ((g_{s})_{s∈I}, f_{i}) = α_{2i}(g_{i}, f_{i}).

Consider the corresponding continuous linear G-birepresentations:

ψ_{i}^{Π} = (ω_{i}: E_{i} × F_{i} → ℝ, α_{1i}^{Π} : G × E_{i} → E_{i}, α_{2i}^{Π} : G × F_{i} → F_{i}).

Then the family {(α_{1i}^{Π})_{x}: G → E_{i}, (α_{2i}^{Π})_{f}: G → F_{i}}_{x∈E_{i},f∈F_{i},i∈I} of all orbit mappings generates the topology τ. Clearly, this implies that (G, τ) is an HBR-group. □

Proposition 4.10. For every family {E_{i}}_{i∈I} of normed spaces, the topological group product ∏ GL(E_{i}) is an HBR-group.

Proof. For each normed space E_{i} from the given family, denote by End(E_{i}) the normed space of all continuous endomorphisms. The action ν_{i}: GL(E_{i}) × End(E_{i}) → End(E_{i}), defined by ν_{i}(A, B) = A·B (multiplication in End(E_{i})), induces a topological group embedding ν̃_{i}: GL(E_{i}) → GL(End(E_{i})). By Remark 4.1, the corresponding representation is bicontinuous. Next, the system {O_{ε}}_{ε>0}, where

O_{ε} = {A ∈ GL(E_{i}): ∥A · Id_{E_{i}} − Id_{E_{i}} ∥ < ε}

is a local base of the natural topology of GL(E_{i}). Hence, GL(E_{i}) carries the strong operator topology with respect to ν̃_{i}. Now, we can use Lemma 4.9 in the situation: E_{i}: = End(E_{i}), F_{i}: = (End(E_{i}))^{∗}, G_{i}: = GL(E_{i}). □

Proposition 4.11. For every locally convex vector space E, the group (E, +) belongs to HBR.

Proof. Since (E, +) is an additive topological subgroup of a product ∏ E_{i} of normed spaces, then by Proposition 3.7 the group ∏ E_{i} is a topological subgroup of the product ∏ GL(ℝ × E_{i}^{∗}). Now our assertion follows from Proposition 4.10. □

Corollary 4.12. Let A(X) be the free Abelian topological group of X. Then A(X) belongs to HBR.

Proof. By a result of Tkačhenko [28], (see also Uspenskii [29, p. 657]), A(X) is a topological subgroup of the free locally convex space L(X) of X. □

The following result solves for Abelian groups (in a slightly stronger form) the above-mentioned problem (Question 2.13(ii)) of Arkhangel’skii.

Theorem 4.13. Every Abelian topological group is a factor group of a perfectly minimal group.

Proof. Every Abelian topological group G is a factor group of its free Abelian topo-

logical group A(G). On the other hand, by Theorem 4.8 and Corollary 4.12, A(G) is a

group retract of a perfectly minimal group. □

Question 4.14. Is every free topological group F (X) a BR-group?

Acknowledgement.

The author thanks the referee for numerous suggestions and corrections which

greatly improved the contents of this article.

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E-mail: megereli@bimacs.cs.biu.ac.il