Extending Topological Abelian Groups by the Unit Circle

Remark 1. Observe that a necessary condition for the splitting of the twisted sum of topological Abelian groups $$ is that i(H) be a dually embedded subgroup of X.

Theorem 2. Let $$ be a twisted sum of topological Abelian groups. The following are equivalent:

• (1)

  $$;

• (2)

  there exists a continuous homomorphism  S : G → X with π∘S = id_G, i.e., a right inverse for  π;

• (3)

  there exists a continuous homomorphism  P : X → H with P∘ı = id_H, i.e., a left inverse for  ı.

Lemma 3. Let K be a compact subgroup of a topological Abelian group X. X^∧ separates points of K if and only if K is dually embedded in X.

Proof. Suppose that X^∧ separates points of K. It is known that for any locally compact Abelian group G, a subgroup L ≤ G^∧ is dense in G^∧ if and only if it separates points of G (see [9, Proposition 31]). The subgroup L of K^∧ formed by all restrictions of characters of X separates points of K by hypothesis. Hence L is dense in K^∧ and, as K^∧ is discrete, L coincides with K^∧.

Suppose that K is dually embedded in X. As K is compact, it is a MAP group. Fix a nonzero x ∈ K. There exists χ ∈ K^∧ such that χ(x) ≠ 1. Since K is dually embedded in X, there exists an extension $$ of χ with $$.

Corollary 4 ([10, Proposition 1.4]). Let K be a compact subgroup of a topological Abelian group X. If X is maximally almost periodic, then K is dually embedded in X.

Theorem 5. Let $$ be a twisted sum of topological Abelian groups. The following are equivalent:

• (1)

  the twisted sum $$ splits;

• (2)

  X^∧ separates points of ı(𝕋);

• (3)

  ı(𝕋) is dually embedded in X.

Proof. 2⇔3 is a corollary of Lemma 3.

3⇒1: suppose that ı(𝕋) is dually embedded in X. Hence there exists a continuous character χ : X → 𝕋 which extends the isomorphism φ : ı(𝕋) → 𝕋 defined by φ(ı(t)) = t. Since χ∘ı = id_𝕋, the assertion follows from Theorem 2.

1⇒2: fix x ∈ ı(𝕋), x = ı(z) with z ≠ 1. By Theorem 2, there exists a continuous homomorphism P : X → 𝕋 with P∘ı = id_𝕋; hence P(ı(z)) = id_𝕋(z) = z ≠ 1.

Lemma 6 ([3, Theorem 2.1]). Let H be a locally quasi-convex subgroup of a topological Abelian group X such that X/H is locally quasi-convex. Then X is locally quasi-convex if and only if H is dually embedded in X.

Theorem 7. Let $$ be a twisted sum of topological Abelian groups. Suppose that G is locally quasi-convex. Then conditions (1), (2), (3) of Theorem 5 are equivalent to

• (4)

  X is locally quasi-convex.

Proof. 1⇒4: if the twisted sum splits, X is topologically isomorphic to the product of two locally quasi-convex groups; hence it is locally quasi-convex.

4⇒3: given a twisted sum $$ with X and G locally quasi-convex, since G≅X/ı(𝕋), by Lemma 6, we deduce that ı(𝕋) is dually embedded in X.

Definition 8. We say that a topological Abelian group G is in the class S _TG (𝕋) if every twisted sum of topological Abelian groups 0 → 𝕋 → X → G → 0 splits.

Theorem 9. Let G be a topological vector space such that G ∈ S _TG (𝕋) as a topological group. Then G is a 𝒦-space.

Proof. Suppose that G ∈ S _TG (𝕋). Fix a twisted sum of topological vector spaces $$. Recall that we denote by p : ℝ → 𝕋 the canonical projection, given by p(r) = exp (2πir). If we consider the push-out PO of p and ı, we obtain a commutative diagram

where the bottom sequence is a twisted sum. Since G ∈ S _TG (𝕋), this sequence splits. Hence there exists a left inverse φ for r. Then φ∘s∘ı = p. Since X is a topological vector space, φ∘s is of the form x ↦ exp (2πif(x)) for some continuous linear functional f ∈ X^*. This clearly implies f∘ı = id_ℝ; that is, f is a left inverse for ı; hence the top sequence splits, too.

Theorem 10 ([4, 12, 13]). There is a short exact sequence of topological vector spaces and continuous, relatively open linear maps $$ which does not split. In other words, ℓ₁ is not a 𝒦-space.

Corollary 11. ℓ₁ ∉ S _TG (𝕋).

Remark 12. The above corollary gives an example of a quotient X/𝕋 which is locally quasi-convex as a topological group but such that X does not even separate points of 𝕋: a strong failure of the 3-space property for local quasi-convexity and for the property of being a MAP group.

Proposition 13. Let G be a topological Abelian group. Suppose that there exists a subgroup A ≤ G which is in S_TG(𝕋) and satisfies the following property. For every twisted sum of topological Abelian groups of the form $$, the subgroup π⁻¹(A) is dually embedded in X. Then $$.

Corollary 14. Let G be a topological Abelian group. Suppose that there exists an open subgroup A ≤ G such that A ∈ S_TG(𝕋). Then G ∈ S_TG(𝕋).

Proof. If A is an open subgroup of G, with the notation of Proposition 13, π⁻¹(A) is an open subgroup of X and hence dually embedded.

Corollary 15. Let G be a topological Abelian group. Suppose that there exists a dense subgroup A ≤ G such that A ∈ S_TG(𝕋). Then G ∈ S_TG(𝕋).

Proof. If A is a dense subgroup of G, with the notation of Proposition 13, π⁻¹(A) is a dense subgroup of X because π is continuous and onto. In particular it is dually embedded in X.

Lemma 16 ([14, Theorem 6.11]). If G is a metrizable topological group that has a completion $$ and if H is a closed normal subgroup of G, then G/H has a completion that is topologically isomorphic to $$, where $$ is the closure of H in $$.

Proposition 17. Let G be a metrizable topological Abelian group which is in $$. Suppose that A is a dense subgroup of G. Then A is in S_TG(𝕋), too.

Proof. Let X be a topological Abelian group, and let 𝕋≅L ≤ X be a subgroup such that A≅X/L. Since metrizability is a 3-space property ([15, 5.38(e)]), X is a metrizable group. By Lemma 16, $$. By Corollary 15, $$. It follows that L is dually embedded in $$; hence it is dually embedded in X, too.

Theorem 18. Locally precompact Hausdorff Abelian groups are in S_TG(𝕋).

Proof. Let G be a locally precompact Hausdorff Abelian group. Given a twisted sum $$, as 𝕋 is in particular precompact, by the above argument X is locally precompact, too. But every subgroup of a locally compact group is dually embedded [15, 24.12], so ı(𝕋) is dually embedded in $$, the completion of X. Then ı(𝕋) is also dually embedded in X and by Theorem 5, the twisted sum splits.

Corollary 19. Locally compact Hausdorff Abelian groups and precompact Hausdorff Abelian groups are in S_TG(𝕋).

Remark 20. It is proved in [11] that every topological vector space endowed with its weak topology is a 𝒦-space. The above corollary shows that a similar result is true for topological Abelian groups, since a topological group endowed with the topology induced by its characters is precompact (see [16]).

Theorem 21. Let G be a topological Abelian group.

• (1)

  If a closed subgroup H ≤ G is such that G/H ∈ S_TG(𝕋), then H is dually embedded.

• (2)

  If G is in S_TG(𝕋) and H ≤ G is a closed dually embedded subgroup, then G/H ∈ S_TG(𝕋).

Proof. (1) Suppose that G/H ∈ S _TG (𝕋). Let χ : H → 𝕋 be a character, and consider the natural twisted sum $$. By taking the corresponding push out, we obtain the following commutative diagram:

where both rows are twisted sums of topological Abelian groups. Since G/H ∈ S _TG (𝕋), the bottom sequence splits; hence by Theorem 2, there exists a continuous homomorphism T : PO → 𝕋 such that T∘r = id_𝕋. The homomorphism T∘s is an extension of χ.

(2) Suppose that H is dually embedded in G. Fix a twisted sum $$. Let q : G → G/H be the canonical projection, and let PB = {(g, x) ∈ G × X : q(g) = π(x)} be the pull-back of q and π. Define r : PB → G and s : PB → X, as the restrictions of the corresponding projections. Note that

We thus obtain the following diagram:

Note that since q and π are onto, the definition of PB yields r((U × X)∩PB)⊃U and s((G × V)∩PB)⊃V for every U ∈ 𝒩₀(G) and V ∈ 𝒩₀(X); hence r and s are onto and open. Thus both short sequences are twisted sums of topological Abelian groups.

Fix φ ∈ ı(𝕋) ^∧; we will find an extension of φ to the whole X. We can regard φ as a character of {0} × ı(𝕋) by defining $$. Since {0} × ı(𝕋)≅𝕋 and G ∈ S _TG (𝕋), by Theorem 5, {0} × ı(𝕋) is dually embedded in PB. Thus there exists ψ ∈ PB^∧ with $$; that is, ψ(0, x) = φ(x) for every x ∈ ı(𝕋). Define, for every h ∈ H, $$ (note that if h ∈ H then (h, 0) ∈ PB).

Since H is dually embedded in G there exists σ ∈ G^∧ with $$ i.e. σ(h) = ψ(h, 0) for every h ∈ H. Now define ρ ∈ PB^∧ as follows: $$. This is clearly continuous. Note that ker   ρ ≥ ker   s = H × {0} since if h ∈ H, we have that $$. As X≅PB/(H × {0}), the character $$ in X^∧ given by $$ for every (g, x) ∈ PB is well defined and continuous.

Now $$ is the desired extension of φ: if x ∈ ı(𝕋), we have $$.

Remark 22. An analogous result in the framework of F-spaces is [1, Theorem 5.2] (cf. also [11, Lemma 4.1]).

Corollary 23. A topological Abelian group G is in S _TG (𝕋) if and only if whenever X is a topological Abelian group and H is a closed subgroup of X with X/H≅G, then H is dually embedded.

Proof. This follows from Theorems 21(1) and 5.

Proposition 24. Let (G_α) _α∈I be a family of topological Abelian groups in S _TG (𝕋). The coproduct ⨁_α∈IG_α is in S _TG (𝕋).

Proof. Let $$ be a twisted sum. Consider, for each β ∈ I, the pull-back PB_β of π and i_β : G_β → ⨁_α∈IG_α; for every β ∈ I there is a commutative diagram

whose rows are twisted sums. As G_β ∈ S _TG (𝕋), by Theorem 2, there exists a homomorphism S_β : G_β → PB_β such that $$. Consider the map As v_α∘S_α is a continuous homomorphism, by the universal property of the coproduct, S is a continuous homomorphism. For every g = (g_α) _α∈I ∈ ⨁_α∈IG_α, we have

so S is a right inverse for π, and again by Theorem 2, the initial twisted sum splits.

The class of nuclear groups was formally introduced by Banaszczyk in [8]. His aim was to find a class of topological groups enclosing both nuclear spaces and locally compact Abelian groups (as natural generalizations of finite-dimensional vector spaces). The original definition is rather technical, as could be expected from its success in gathering objects from such different classes into the same framework. Next we collect some facts concerning the class of nuclear groups which are relevant to this paper.

• (i)

  Nuclear groups are locally quasi-convex [8, 8.5].

• (ii)

  Subgroups of nuclear groups are dually embedded [8, 8.3].

• (iii)

  Products, countable coproducts, subgroups, and Hausdorff quotients of nuclear groups are nuclear [8, 7.6, 7.8, 7.5].

• (iv)

  Every locally compact Abelian group is nuclear [8, 7.10].

• (v)

  A nuclear locally convex space is a nuclear group [8, 7.4]. Furthermore, if a topological vector space is a nuclear group, then it is a locally convex nuclear space [8, 8.9].

Theorem 25. Let $$ be a countable direct system of nuclear Abelian groups in S _TG (𝕋). Then the direct limit lim _→ G_n is in S _TG (𝕋). In particular, sequential direct limits of locally compact groups are in S _TG (𝕋).

Proof. The direct sum $$ with the coproduct topology is in S _TG (𝕋) by Proposition 24. Let i_m : G_m → ⨁G_n be the inclusion map, for every m ∈ ℕ. It is known (see [17]) that $$, where $$ is the closure of the subgroup H generated by $$. Since countable coproducts of nuclear groups are nuclear groups, $$ is dually embedded. By Theorem 21, lim _→ G_n ∈ S _TG (𝕋).

Varopoulos introduced in [18] the class ℒ_∞ of all topological groups whose topologies are the intersection of a decreasing sequence of locally compact Hausdorff group topologies. He succeeded in his aim of extending known results about locally compact groups and established the basis for the development of the harmonic analysis on ℒ_∞ groups. Subsequently many other authors investigated different properties of this class ([19–23]).

The following is a relevant fact concerning the structure of ℒ_∞ groups proved by Sulley.

Proposition 26 ([22]). Let G be any Abelian group endowed with an ℒ_∞ topology. Then G has an open subgroup which is a strict inductive limit of a sequence of Hausdorff locally compact Abelian groups.

Corollary 27. Let G be any Abelian group endowed with an ℒ_∞ topology. Then G is in S _TG (𝕋).

Proof. By the above proposition and Theorem 25, G has an open subgroup in S _TG (𝕋). Hence Corollary 14 implies that G is in S _TG (𝕋).

Definition 28 ([7]). Let G and H be topological Abelian groups and ω : G → H a map with ω(0) = 0. We say that ω is a quasi-homomorphism if the map Δ_ω : (x, y) ∈ G × G ↦ ω(x + y) − ω(x) − ω(y) ∈ H is continuous at (0,0).

A quasi-homomorphism ω : G → H is approximable if there exists a homomorphism a : G → H such that ω − a is continuous at 0.

Proposition 29. Let G and H be topological Abelian groups and ω : G → H a quasi-homomorphism.

• (1)

  The sets

  $$ ()

•  

  form a basis of neighborhoods of zero for a group topology on H × G.

• (2)

  If H⨁_ωG denotes the group H × G endowed with the topology induced by the quasi-homomorphism ω and ı_H and π_G denote the canonical inclusion and projection, respectively, $$ is a twisted sum of topological Abelian groups.

• (3)

  A quasi-homomorphism is approximable if and only if the induced twisted sum splits.

• (4)

  The twisted sum $$ is equivalent to one induced by a quasi-homomorphism if and only if it splits algebraically and there exists a map ρ : G → X such that π∘ρ = id_G, ρ(0) = 0 and ρ is continuous at the origin.

Lemma 30. Let G and H be topological Abelian groups and ω : G → H a quasi-homomorphism. Then the map g ∈ G ↦ (ω(g), g) ∈ H⨁_ωG is continuous at 0.

Proof. Note that for every U ∈ 𝒩₀(G) and V ∈ 𝒩₀(H) we have [g ∈ U⇒(ω(g), g) ∈ W(V, U)].

Proposition 31. Let π : X → G be a continuous and open surjective homomorphism between topological Abelian groups. Suppose that X is metrizable. Then there exists ρ : G → X such that π∘ρ = id_G and ρ is continuous at 0 ∈ G.

Proof. Note that in order to define ρ with π∘ρ = id_G, we simply must choose for every g ∈ G an element x ∈ π⁻¹(g), which is a nonempty set since π is onto. Let us see that it can be done in such a way that the map thus obtained is continuous at zero.

Let {U_n : n ∈ ℕ} be a decreasing basic sequence of neighborhoods of zero in X, where U₁ = X. Due to the continuity of π, we have ⋂_n∈ℕ π(U_n) = {0}. Let ρ take the value 0 on g = 0. For any g ≠ 0, by the previous paragraph, we can choose n and x with π(x) = g,  x ∈ U_n,  g ∉ π(U_n+1), and define ρ(g) = x. Now fix m ∈ ℕ; we must find V ∈ 𝒩₀(G) with ρ(V)⊆U_m. Since π is open there exists V ∈ 𝒩₀(G) with π(U_m)⊇V. Fix g ∈ V, and let us show that ρ(g) ∈ U_m. If ρ(g) = 0 this is trivial. Otherwise ρ(g) = x with π(x) = g,    x ∈ U_n,  g ∉ π(U_n+1) for some n. Then g ∈ V⊆π(U_m); hence m ≤ n and x ∈ U_n⊆U_m.

Corollary 32. Let G be a metrizable topological Abelian group.

• (1)

  If H is metrizable and divisible, every twisted sum $$ is equivalent to one induced by a quasi-homomorphism.

• (2)

  G is in S _TG (𝕋) if and only if every quasi-character ω : G → 𝕋 is approximable.

Proof. Metrizability is a 3-space property (see [15, 5.38(e)]). If $$ is a twisted sum where both G and H are metrizable, so is X, and thus the hypotheses of Proposition 31 hold. Therefore, there exists a section of π continuous at the origin. As H is a divisible group, the twisted sum $$ splits algebraically. By Proposition 29(4), this twisted sum is equivalent to one induced by a quasi-homomorphism.

The second part is a consequence of the first one and Proposition 29(3).

Definition 33. For any t ∈ 𝕋 let us call 〈t〉 the only real number α ∈   ] − 1/2,1/2] such that exp (2πiα) = t. Then [t→|〈t〉|] is a norm on 𝕋. (Note that 〈ts〉∈〈t〉+〈s〉+ℤ; hence |〈ts〉|≤|〈t〉+〈s〉|≤|〈t〉|+|〈s〉.) Put θ(t) = |〈t〉|. For every β > 0 let us define T_β = {t ∈ 𝕋 : θ(t) ≤ β}. Note that 𝕋₊ = T_1/4 with this notation.

Definition 34. For every Abelian group G, every V⊆G with 0 ∈ V, and every n ∈ ℕ we define (1/2ⁿ)V = {x ∈ V   :   2^kx ∈ V  ∀ k ∈ {0,1, …, n}}.

Proposition 35 ([24, Corollary 2]). For every n ∈ ℕ ∪ {0} and β ∈ [0,1/3), we have $$.

Lemma 36. Let G be a topological Abelian group, and let ω : G → 𝕋 be a quasi-character. Suppose that there exist U ∈ 𝒩₀(G) and β ∈ (0,1/3) such that ω(U)⊆T_β. Then ω is continuous at zero.

Proof. Since ω is a quasi-character, for every ρ > 0, there exists W_ρ ∈ 𝒩₀(G) with $$ for every u ∈ W_ρ. Fix any ɛ > 0. Let us find V ∈ 𝒩₀(G) with ω(V)⊆T_ɛ.

Fix any β^′ ∈ (β, 1/3). Find N ∈ ℕ with β^′/2^N ≤ ɛ. Put $$.

It is enough to prove that

since by Proposition 35, this will imply $$. Now, for every n ∈ {0,1, 2, …, N}

Since v ∈ (1/2^N)U, we have that ω(2ⁿv) ∈ T_β. Now, for every j ∈ {0, …, n − 1} and every n ∈ {0, …, N}, $$, since $$. Thus $$, and we deduce $$. This implies that $$.

Corollary 37. A quasi-character ω : G → 𝕋 is approximable if and only if there exist U ∈ 𝒩₀(G), β ∈ (0,1/3) and an algebraic character χ ∈ Hom (G, 𝕋) such that $$. Moreover any such χ approximates ω.

Lemma 38 ([7, Lemma 6]). Let G be an Abelian group (no topology is assumed), and let ω : G → 𝕋 be any mapping such that for some β < 1/3, $$. Then there is a unique character χ : G → 𝕋 such that $$.

Theorem 39. Let μ be a nonatomic σ-finite measure on a set Δ. Let L₀ : = L₀(μ) be the space of all measurable functions on Δ with the norm

Then every quasi-character ω : L₀ → 𝕋 is approximable.

Proof. We can assume, without loss of generality, that μ is a probability (note that L₀(μ) is topologically isomorphic to L₀(ν), where ν is a probability with the same null sets as μ).

Let ω : L₀ → 𝕋 be a quasi-character, fix β < 1/3, and choose δ₀ such that $$ for every f, g with ∥f∥₀, ∥g∥₀ ≤ δ₀.

Let $$ be a partition of Δ into measurable sets, with μ(Δ_i) ≤ δ₀ for all 1 ≤ i ≤ r. Then $$ as a topological direct product. For all f ∈ L₀(Δ_i) we have that

Call ω_i the restriction of ω to each L₀(Δ_i). As $$ for every 1 ≤ i ≤ r and f, g ∈ L₀(Δ_i), we can apply Lemma 38 to obtain unique characters χ_i : L₀(Δ_i) → 𝕋 such that $$ for every f ∈ L₀(Δ_i). By Lemma 36, we have that each $$ is continuous at the origin of L₀(Δ_i), and, thus, the character χ : L₀ → 𝕋 given by $$ (where $$) approximates ω near the origin.

Corollary 40. L₀ is in S _TG (𝕋).

Proof. This follows from Theorem 39 and Corollary 32, since L₀ is a metrizable group.

Example 41. Let L₀ be as in Theorem 39. Fix a discrete, nontrivial D ≤ L₀ (e.g., a copy of ℤ). Note that L₀ does not have any nontrivial continuous character, and in particular D is not dually embedded in L₀. Using Theorem 21 we deduce that L₀/D is not in S _TG (𝕋). Since L₀ ∈ S _TG (𝕋), this example shows that being in S _TG (𝕋) is not preserved by local isomorphisms (compare with Corollary 14).

Proposition 42. Let G be a protodiscrete topological Abelian group. Every quasi-character of G is approximable.

Proof. Let ω : G → 𝕋 be a quasi-character. There exists an open subgroup U ≤ G such that $$ for every a, b ∈ U. Using Lemma 38 we deduce that there exists an algebraic character χ : U → 𝕋 with $$ for every u ∈ U. Now Corollary 37 implies that any algebraic extension of χ approximates ω.

Corollary 43. Every protodiscrete, metrizable group is in S _TG (𝕋).

Proof. This follows from Proposition 42 and Corollary 32.

Example 44. Countable products of discrete Abelian groups belong to S _TG (𝕋).