Extending Topological Abelian Groups by the Unit Circle

Abstract

A twisted sum in the category of topological Abelian groups is a short exact sequence 0→𝑌→𝑋→𝑍→0 where all maps are assumed to be continuous and open onto their images. The twisted sum splits if it is equivalent to 0→𝑌→𝑌×𝑍→𝑍→0. We study the class 𝑆_TG(𝕋) of topological groups G for which every twisted sum 0→𝕋→𝑋→𝐺→0 splits. We prove that this class contains Hausdorff locally precompact groups, sequential direct limits of locally compact groups, and topological groups with ℒ_∞ topologies. We also prove that it is closed by taking open and dense subgroups, quotients by dually embedded subgroups, and coproducts. As means to find further subclasses of 𝑆_TG(𝕋), we use the connection between extensions of the form 0→𝕋→𝑋→𝐺→0 and quasi-characters on G, as well as three-space problems for topological groups. The subject is inspired on some concepts known in the framework of topological vector spaces such as the notion of 𝒦-space, which were interpreted for topological groups by Cabello.

1. Introduction and Preliminaries

In the theory of topological vector spaces (topological groups) a property 𝑃 is said to be a 3-space property if whenever a closed subspace (subgroup) 𝑌 of a space (group) 𝑋 and the corresponding quotient 𝑋/𝑌 both have property 𝑃, 𝑋 also has property 𝑃.

A short exact sequence of topological vector spaces (topological groups) 0→𝑌→−^𝚤𝑋−→^𝜋𝑍→0 will be called a twisted sum, and the space (group) 𝑋 will be called an extension of 𝑍 by 𝑌 when both 𝚤 and 𝜋 are continuous and open onto their images. Using this language, 3-space properties can be described as those which are preserved by forming extensions.

An example of a 3-space property in the category of Banach spaces is reflexivity. However, the point-separating property (i.e., having a dual space which separates points) is not a 3-space property. (Consider the space 𝑙_𝑝 for 0<𝑝<1, and a weakly closed subspace 𝑀 of 𝑙_𝑝 without the Hahn-Banach extension property. If we take as 𝑁 the kernel of some continuous linear functional on 𝑀 which does not extend to 𝑙_𝑝, then 𝑋=𝑙_𝑝/𝑁 does not have the point-separating property but both 𝐿∶=𝑀/𝑁 and 𝑋/𝐿 have this property (see [1])).

In the category of topological Abelian groups, local compactness, precompactness, metrizability, and completeness are 3-space properties. However, 𝜎-compactness, sequential completeness, realcompactness, and a number of other properties are not (see [2] for more examples).

The twisted sum 0→𝑌→−^𝚤𝑋−→^𝜋𝑍→0 splits if there exists a continuous linear map (continuous homomorphism) 𝑇∶𝑋→𝑌×𝑍 making the following diagram commutative (𝚤_𝑌 is the canonical inclusion of 𝑌 into the product, and 𝜋_𝑍 is the canonical projection onto 𝑍): (1)

It is known that if such a 𝑇 exists, it must actually be a topological isomorphism.

If the twisted sum 0→𝑌→−^𝚤𝑋−→^𝜋𝑍→0 splits and both 𝑌 and 𝑍 have a productive property 𝑃, then 𝑋 has property 𝑃, too.

Kalton et al. provided in [1] the first formal and extensive study of splitting twisted sums in the framework of 𝐹-spaces (complete metric linear spaces). They devote Chapter 5 of this monograph to the following problem: is local convexity a 3-space property?

On the path to answering (in the negative) this question, the authors mention a useful result by Dierolf (1973) which asserts that there exists a non-locally-convex extension of the 𝐹-space 𝑍 by the 𝐹-space 𝑌 if and only if there exists a non-locally-convex extension of 𝑍 by ℝ. The analogue of this result for topological groups, which involves the notion of local quasi-convexity, was obtained by Castillo in [3].

At this point the following definition, originally introduced in [4], comes across as natural: a 𝐹-space 𝑋 is said to be a 𝒦-space if, whenever 𝑌 is an 𝐹-space and 𝐿 is a subspace of 𝑌 with dimension one such that 𝑌/𝐿≅𝑋, the corresponding twisted sum splits. The negative answer to the 3-space problem for local convexity is obtained in [1] as a corollary of the fact that ℓ₁ is not a 𝒦-space.

The notion of 𝒦-space is relevant on its own, regardless of 3-space properties. Many classical spaces such as 𝐿_𝑝 (0<𝑝<∞), 𝑙_𝑝 (𝑝≠1), or 𝑐₀ are 𝒦-spaces.

In this paper we will study the natural counterpart of the notion of 𝒦-space for topological groups and its connections with 3-space problems, following the work started by Cabello in [5–7].

For simplicity, and because our methods are applicable for the most part only to Abelian groups, we use additive notation, and denote by 0 the neutral element. We denote by ℕ the set of all positive natural numbers, by ℤ the integers, by ℝ the reals, by ℂ the set of complex numbers, and by 𝕋 the unit circle of ℂ, with the topology induced by ℂ. In 𝕋 we will use multiplicative notation and we will denote by 𝑝 the canonical projection from ℝ to 𝕋 given by 𝑝(𝑡)=exp(2𝜋𝑖𝑡). We will use 𝒩₀(𝐺) to denote the system of neighborhoods of 0 in a topological Abelian group 𝐺.

Recall that a topological Abelian group 𝐺 is precompact if for every neighborhood of zero 𝑉 there exists a finite subset 𝐹 of 𝐺 such that 𝐺=𝐹+𝑉. Precompact groups are the subgroups of compact groups. In the same way a group 𝐺 is locally precompact if and only if it is a subgroup of a locally compact group.

The dual group  𝐺^∧ of a given topological Abelian group 𝐺 is formed from the continuous group homomorphisms from 𝐺 into 𝕋, usually called characters. The polar set of a subset 𝐴 of 𝐺 is defined by 𝐴⁰∶={𝜒∈𝐺^∧∶𝜒(𝐴)⊆𝕋₊}, where 𝕋₊∶=𝑝([−1/4,1/4])={𝑧∈𝕋∶Re(𝑧)≥0}. A subset 𝐴 of a topological group 𝐺 is called quasi-convex if for every 𝑥∈𝐺⧵𝐴 there is a 𝜒∈𝐴⁰ such that 𝜒(𝑥)∉𝕋₊. A topological group is called locally quasi-convex if it has a neighborhood basis of 0 consisting of quasi-convex sets. It is well known (see [8]) that a topological vector space is locally convex if and only if it is a locally quasi-convex topological group in its additive structure.

If 𝐺^∧ separates the points of 𝐺 we say that 𝐺 is maximally almost periodic (MAP). Every locally quasi–convex group is a MAP group.

The analogous notion to the HBEP (Hahn-Banach Extension Property) for topological groups is the following. A subgroup 𝐻≤𝑋 is dually embedded in 𝑋 if each character of 𝐻 can be extended to a character of 𝑋.

When endowed with the compact-open topology 𝜏_co, 𝐺^∧ becomes a Hausdorff topological group. A basis of neighborhoods of the neutral element for the compact open topology 𝜏_co is given by the sets 𝐾⁰={𝜒∈𝐺^∧∶𝜒(𝐾)⊆𝕋₊}, where 𝐾 is a compact subset of 𝐺.

Remark 1. Observe that a necessary condition for the splitting of the twisted sum of topological Abelian groups 0→𝐻→−^𝚤𝑋−→^𝜋𝐺→0 is that 𝑖(𝐻) be a dually embedded subgroup of 𝑋.

The following known characterization is essential when dealing with twisted sums in different categories (see [3, Lemma 3.1] for a proof).

Theorem 2. Let 0→𝐻→−^𝚤𝑋−→^𝜋𝐺→0 be a twisted sum of topological Abelian groups. The following are equivalent: (1)0→𝐻→−^𝚤𝑋−→^𝜋𝐺→0𝑠𝑝𝑙𝑖𝑡𝑠; (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.

We will use the notions of pull-back and push-out in the category of topological Abelian groups, following Castillo [3]. Given topological Abelian groups 𝐴,  𝐵, and 𝐶 and continuous homomorphisms 𝑢∶𝐴→𝐵 and 𝑣∶𝐴→𝐶, the push-out of 𝑢 and 𝑣 is a topological group PO and two continuous homomorphisms 𝑟 and 𝑠 making the square diagram commutative (2)

and such that for every topological Abelian group PO^′ and continuous homomorphisms 𝑟^′∶𝐶→PO^′ and 𝑠^′∶𝐵→PO^′ with 𝑠^′∘𝑢=𝑟^′∘𝑣, there is a unique continuous homomorphism 𝜙 from PO to PO^′ making the two triangles commutative. The topological group PO exits and is unique up to topological isomorphism.

Given any twisted sum 0→𝐻→−^𝚤𝑋−→^𝜋𝐺→0 of topological Abelian groups, any topological Abelian group 𝑌, and any continuous homomorphism 𝑡∶𝐻→𝑌, if PO is the push-out of 𝚤 and 𝑡, there is a commutative diagram (3)

where both squares are commutative and the bottom sequence is a twisted sum [3].

An analogous result for the dual construction (the pull-back) can be obtained (see [3]).

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 𝜒∈𝐺^∧ of 𝜒 with 𝜒(𝑥)=𝜒(𝑥)≠1.

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.

2. The Class 𝑆_TG(𝕋)

Next we consider the particular case in which the compact subgroup 𝐾 is 𝕋. (The proofs of Theorems 5 and 7 below can be extracted from that of [3, Theorem 4.1], but we prefer the present formulation.)

Theorem 5. Let 0→𝕋→−^𝚤𝑋−→^𝜋𝐺→0 be a twisted sum of topological Abelian groups. The following are equivalent: (1)the twisted sum 0→𝕋→−^𝚤𝑋−→^𝜋𝐺→0 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.

Now we will complete the previous Theorem under the assumption that 𝐺 is locally quasi-convex. We will use the following result due to Castillo, concerning the 3-space problem in locally quasi-convex groups.

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 0→𝕋→−^𝚤𝑋−→^𝜋𝐺→0 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 0→𝕋→−^𝚤𝑋−→^𝜋𝐺→0 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.

Following the notation used by Domański [11] in the framework of topological vector spaces, we introduce the class 𝑆_TG(𝕋) which is the analogue of that of 𝒦-spaces for topological Abelian groups.

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 0→ℝ→−^𝚤𝑋−→^𝜋𝐺→0. 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 (4)
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 0→ℝ→−^𝚤𝑋−→^𝜋𝐺→0 which does not split. In other words, ℓ₁ is not a 𝒦-space.

Using Theorem 9 we deduce the following.

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.

From Theorem 5 it follows that a topological Abelian group 𝐺 is in 𝑆_TG(𝕋) iff for every twisted sum of topological Abelian groups of the form 0→𝕋→−^𝚤𝑋−→^𝜋𝐺→0, the subgroup 𝚤(𝕋) is dually embedded in 𝑋. Note that in any such twisted sum, for an arbitrary 𝐴≤𝐺 the subgroup 𝜋⁻¹(𝐴) contains 𝚤(𝕋). This yields the following criterion;

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 0→𝕋→−^𝚤𝑋−→^𝜋𝐺→0, the subgroup 𝜋⁻¹(𝐴) is dually embedded in 𝑋. Then 𝐺∈𝑆_TG(𝕋).

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.

We next see that the converse of Corollary 15 is true in the case of 𝐺 metrizable. For any topological Abelian group 𝐺, we denote by 𝐺 the Raĭkov completion of 𝐺. See [14] for more information about this subject.

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

Proposition 17. Let 𝐺 be a metrizable topological Abelian group which is in 𝑆_TG(𝕋). 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, 𝐺=𝐴≅𝑋/𝐿. By Corollary 15, 𝐺∈𝑆_TG(𝕋). It follows that 𝐿 is dually embedded in 𝑋; hence it is dually embedded in 𝑋, too.

Our next aim is to prove that Hausdoff locally precompact groups are in 𝑆_TG(𝕋). Note first that if 𝐺 is a topological Abelian group and 𝐻≤𝐺 is a precompact subgroup such that the quotient 𝐺/𝐻 is locally precompact, then 𝐺 is locally precompact, too. (Indeed, let 𝜋∶𝐺→𝐺/𝐻 be the canonical projection. Choose 𝑈∈𝒩₀(𝐺) such that 𝜋(𝑈) is precompact. Let us see that 𝑈 is precompact. Given 𝑉∈𝒩₀(𝐺) we need to find a finite subset 𝐹⊂𝐺 with 𝑈⊂𝐹+𝑉. Fix 𝑉^′∈𝒩₀(𝐺) with 𝑉^′+𝑉^′⊂𝑉 and find a finite 𝐹₁⊂𝐺 with 𝐻⊂𝐹₁+𝑉^′. Since 𝜋(𝑈) is precompact there exists a finite subset 𝐹^′₂ of 𝐺/𝐻 with 𝜋(𝑈)⊂𝐹^′₂+𝜋(𝑉^′). We may suppose that 𝐹^′₂=𝜋(𝐹₂) with 𝐹₂ being a finite subset of 𝐺. Hence 𝜋(𝑈)⊂𝜋(𝐹₂)+𝜋(𝑉^′)=𝜋(𝐹₂+𝑉^′), which implies 𝑈⊂𝜋⁻¹𝜋(𝑈)⊂𝜋⁻¹𝜋(𝐹₂+𝑉^′)=𝐹₂+𝑉^′+𝐻⊂𝐹₁+𝐹₂+𝑉^′+𝑉^′⊂𝐹+𝑉 if we put 𝐹=𝐹₁+𝐹₂.)

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 0→𝕋→−^𝚤𝑋−→^𝜋𝐺→0, 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 𝑋, 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 0→𝐻→−^𝚤𝐺−→^𝜋𝐺/𝐻→0. By taking the corresponding push out, we obtain the following commutative diagram:(5)
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 0→𝕋→−^𝚤𝑋−→^𝜋𝐺/𝐻→0. 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 ==)(𝑥={0}(𝕋),==𝑔{0}.ker𝑟=𝑔,𝑥∈PB∶𝑟𝑔,𝑥=0𝑔,𝑥∈PB∶𝑔=0{(0,𝑥∈𝐺×𝑋∶𝐻=𝜋)}×𝚤ker𝑠=𝑔,𝑥∈PB∶𝑠𝑔,𝑥=0𝑔,𝑥∈PB∶𝑥=0𝑔,0∈𝐺×𝑋∶𝑞=𝐻=𝐻× (6)
We thus obtain the following diagram: (7)
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 𝜑(0,𝑥)=𝜑(𝑥). 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 𝜓|_{{0}×𝚤(𝕋)}=𝜑; that is, 𝜓(0,𝑥)=𝜑(𝑥) for every 𝑥∈𝚤(𝕋). Define, for every ℎ∈𝐻, 𝜓(ℎ)=𝜓(ℎ,0) (note that if ℎ∈𝐻 then (ℎ,0)∈PB).
Since 𝐻 is dually embedded in 𝐺 there exists 𝜎∈𝐺^∧ with 𝜎_|𝐻=𝜓 i.e. 𝜎(ℎ)=𝜓(ℎ,0) for every ℎ∈𝐻. Now define 𝜌∈PB^∧ as follows: 𝜌(𝑔,𝑥)=𝜓(𝑔,𝑥)𝜎(𝑔). This is clearly continuous. Note that ker𝜌≥ker𝑠=𝐻×{0} since if ℎ∈𝐻, we have that 𝜌(ℎ,0)=𝜓(ℎ,0)𝜎(ℎ)=1. As 𝑋≅PB/(𝐻×{0}), the character 𝜌 in 𝑋^∧ given by 𝜌(𝑥)=𝜌(𝑔,𝑥) for every (𝑔,𝑥)∈PB is well defined and continuous.
Now 𝜌 is the desired extension of 𝜑: if 𝑥∈𝚤(𝕋), we have 𝜌(𝑥)=𝜌(0,𝑥)=𝜓(0,𝑥)𝜎(0)=𝜓(0,𝑥)=𝜑(𝑥).

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

The following corollary is a generalization of Theorems 5.2 and 5.3 in [4] and appears as Theorem 4.1 in [3].

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.

Let (𝐺_𝛼)_𝛼∈𝐼 be a family of topological Abelian groups. The coproduct of (𝐺_𝛼)_𝛼∈𝐼 is the direct sum ⨁_𝛼∈𝐼𝐺_𝛼 endowed with the finest group topology making the inclusion maps 𝑖_𝛽∶𝐺_𝛽↪⨁_𝛼∈𝐼𝐺_𝛼 continuous, for every 𝛽∈𝐼. If (𝐺_𝑛)^∞_𝑛=1 is a countable family of groups, this topology coincides with the box topology on ⨁_𝑛∈ℕ𝐺_𝑛.

Recall that the coproduct ⨁_𝛼∈𝐼𝐺_𝛼 has the following universal property. Given an arbitrary topological Abelian group 𝐺 and a homomorphism 𝑓∶⨁_𝛼∈𝐼𝐺_𝛼→𝐺, 𝑓 is continuous if and only if 𝑓∘𝑖_𝛽 is continuous ∀𝛽∈𝐼.

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 0→𝕋→−^𝚤𝑋−→^𝜋⨁_𝛼∈𝐼𝐺_𝛼→0 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 (8)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 𝜋_𝛽∘𝑆_𝛽=id_𝐺𝛽. Consider the map 𝑆∶_𝛼∈𝐼𝐺_𝛼𝑔⟶𝑋_{𝛼𝛼∈𝐼}↦_𝛼∈𝐼𝑣_𝛼∘𝑆_𝛼𝑔_𝛼. (9) 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 𝜋𝑆𝑔=𝜋_𝛼∈𝐼𝑣_𝛼∘𝑆_𝛼𝑔_𝛼=_𝛼∈𝐼𝜋∘𝑣_𝛼∘𝑆_𝛼𝑔_𝛼=_𝛼∈𝐼𝑖_𝛼𝑔_𝛼=𝑔, (10)
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 {𝐺_𝑛,𝑓^𝑚_𝑛}(𝑛≤𝑚) 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 ⨁^∞_𝑛=1𝐺_𝑛 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 lim_→𝐺_𝑛≅(⨁𝐺_𝑛)/𝐻, where 𝐻 is the closure of the subgroup 𝐻 generated by {𝑖_𝑚∘𝑓^𝑚_𝑛(𝑔_𝑛)−𝑖_𝑛(𝑔_𝑛)∶𝑛≤𝑚;𝑔_𝑛∈𝐺_𝑛}. Since countable coproducts of nuclear groups are nuclear groups, 𝐻 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).

3. Quasi-Homomorphisms

In his study of the stability of homomorphisms between topological Abelian groups [7], Cabello defined the notion of quasi-homomorphism, which is inspired by the technique of quasi-linear maps introduced by Kalton and others (see [1]).

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.

Our aim is to use the notion of approximable quasi-homomorphisms to obtain new examples of groups in 𝑆_TG(𝕋).

We start with some facts about quasi-homomorphism taken from [7].

Proposition 29. Let 𝐺 and 𝐻 be topological Abelian groups and 𝜔∶𝐺→𝐻 a quasi-homomorphism. (1) The sets 𝑊(𝑉,𝑈)=ℎ,𝑔∈𝐻×𝐺∶𝑔∈𝑈,ℎ∈𝜔𝑔+𝑉𝑈∈𝒩₀(𝐺),𝑉∈𝒩₀(𝐻) (11) 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, 0→𝐻−−→^𝚤𝐻𝐻⨁_𝜔𝐺−−→^𝜋𝐺𝐺→0 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 0→𝐻→−^𝚤𝑋−→^𝜋𝐺→0 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)].

The following result is [7, Lemma 11]. We give here a proof for the sake of completeness.

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∈ℕ𝜋(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 0→𝐻→−^𝚤𝑋−→^𝜋𝐺→0 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 0→𝐻→−^𝚤𝑋−→^𝜋𝐺→0 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 0→𝐻→−^𝚤𝑋−→^𝜋𝐺→0 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).

We call a norm on an Abelian group 𝐺 any subadditive, symmetric functional ‖⋅‖∶𝐺→[0,∞) such that ‖𝑥‖=0 if and only if 𝑥=0.

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 (1/2^𝑛)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 (1/2^𝑛)(𝑇_𝛽)=𝑇_𝛽/2^𝑛.

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 𝜔(𝑢)²𝜔(2𝑢)∈𝑇_𝜌 for every 𝑢∈𝑊_𝜌. Fix any 𝜀>0. Let us find 𝑉∈𝒩₀(𝐺) with 𝜔(𝑉)⊆𝑇_𝜀.
Fix any 𝛽^′∈(𝛽,1/3). Find 𝑁∈ℕ with 𝛽^′/2^𝑁≤𝜀. Put 𝑉=(1/2^𝑁)𝑈∩(1/2^𝑁−1)𝑊_{(𝛽′−𝛽)/(𝑁2𝑁−1)}.
It is enough to prove that ∀𝑣∈𝑉∀𝑛∈{0,1,2,…,𝑁},𝜔(𝑣)^2𝑛∈𝑇_𝛽′ (12)
since by Proposition 35, this will imply 𝜔(𝑣)∈𝑇_{𝛽′/2𝑁}⊆𝑇_𝜀. Now, for every 𝑛∈{0,1,2,…,𝑁}𝜔(𝑣)^2𝑛=𝜔(𝑣)^2𝑛𝜔(2𝑣)^2𝑛−1𝜔(2𝑣)^2𝑛−1=𝜔(𝑣)²𝜔(2𝑣)^2𝑛−1𝜔(2𝑣)^2𝑛−1=𝜔(𝑣)²𝜔(2𝑣)^2𝑛−1𝜔(2𝑣)^2𝑛−1𝜔(2⋅2𝑣)^2𝑛−2𝜔(2⋅2𝑣)^2𝑛−2=𝜔(𝑣)²𝜔(2𝑣)^2𝑛−1𝜔(2𝑣)²𝜔(2⋅2𝑣)^2𝑛−2𝜔(2⋅2𝑣)^2𝑛−2==⋯_{𝑛−1𝑗=0}𝜔2^𝑗𝑣²𝜔2⋅2^𝑗𝑣^{2𝑛−𝑗−1}𝜔2^𝑛𝑣. (13) Since 𝑣∈(1/2^𝑁)𝑈, we have that 𝜔(2^𝑛𝑣)∈𝑇_𝛽. Now, for every 𝑗∈{0,…,𝑛−1} and every 𝑛∈{0,…,𝑁}, 𝜔(2^𝑗𝑣)²𝜔(2⋅2^𝑗𝑣)∈𝑇_{(𝛽′−𝛽)/(𝑁2𝑁−1)}, since 2^𝑗𝑣∈𝑊_{(𝛽′−𝛽)/(𝑁2𝑁−1)}. Thus (𝜔(2^𝑗𝑣)²𝜔(2⋅2^𝑗𝑣))^{2𝑛−𝑗−1}∈𝑇_{(𝛽′−𝛽)/𝑁}, and we deduce ∏^𝑛−1_𝑗=0(𝜔(2^𝑗𝑣)²𝜔(2⋅2^𝑗𝑣))^{2𝑛−𝑗−1}∈𝑇_{𝛽′−𝛽}. This implies that 𝜔(𝑣)^2𝑛∈𝑇_𝛽′.

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 (𝜔𝜒)(𝑈)⊆𝑇_𝛽. 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, 𝜔(𝑥+𝑦)𝜔(𝑥)𝜔(𝑦)∈𝑇_𝛽∀𝑥,𝑦∈𝐺. Then there is a unique character 𝜒∶𝐺→𝕋 such that 𝜔(𝑥)𝜒(𝑥)∈𝑇_𝛽∀𝑥∈𝐺.

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 𝑓₀∶=_Δmin1,𝑓(𝑡)𝑑𝜇(𝑥). (14) 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 𝜔(𝑓+𝑔)𝜔(𝑓)𝜔(𝑔)∈𝑇_𝛽 for every 𝑓,𝑔 with ‖𝑓‖₀,‖𝑔‖₀≤𝛿₀.
Let Δ=⨁^𝑟_𝑖=1Δ_𝑖 be a partition of Δ into measurable sets, with 𝜇(Δ_𝑖)≤𝛿₀ for all 1≤𝑖≤𝑟. Then 𝐿₀=∏^𝑟_𝑖=1𝐿₀(Δ_𝑖) as a topological direct product. For all 𝑓∈𝐿₀(Δ_𝑖) we have that 𝑓₀=_Δmin1,𝑓(𝑡)𝑑𝜇(𝑥)≤_{(𝑡)𝑡∈Δ∶𝑓≠0}(𝑥)(𝑡)Δ1𝑑𝜇=𝜇𝑡∈Δ∶𝑓≠0≤𝜇_𝑖≤𝛿₀. (15) Call 𝜔_𝑖 the restriction of 𝜔 to each 𝐿₀(Δ_𝑖). As 𝜔_𝑖(𝑓+𝑔)𝜔_𝑖(𝑓)𝜔_𝑖(𝑔)∈𝑇_𝛽 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 𝜔_𝑖(𝑓)𝜒_𝑖(𝑓)∈𝑇_𝛽 for every 𝑓∈𝐿₀(Δ_𝑖). By Lemma 36, we have that each 𝜔_𝑖𝜒_𝑖 is continuous at the origin of 𝐿₀(Δ_𝑖), and, thus, the character 𝜒∶𝐿₀→𝕋 given by 𝜒(𝑓)=∏^𝑟_𝑖=1𝜒_𝑖(𝑓_𝑖) (where 𝑓=∑^𝑟_𝑖=1𝑓_𝑖,𝑓_𝑖∈𝐿₀(Δ_𝑖)) 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).

Let (𝐺,𝜏) be a topological group. We say that (𝐺,𝜏) is a protodiscrete group (or that the topology 𝜏 is linear) if it has a basis of neighborhoods of 0 formed by open subgroups. Note that protodiscrete Hausdorff groups are exactly the subgroups of products of discrete groups.

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 𝜔(𝑎+𝑏)𝜔(𝑎)𝜔(𝑏)∈𝕋₊ for every 𝑎,𝑏∈𝑈. Using Lemma 38 we deduce that there exists an algebraic character 𝜒∶𝑈→𝕋 with 𝜔(𝑢)𝜒(𝑢)∈𝕋₊ 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).

Acknowledgments

The authors acknowledge the financial support of Spanish MICINN (Grant MTM2009-14409-C02-01), and of University of Navarra (PIUNA Grant). The third listed author acknowledges support of Consellería de Educación e Ordenación Universitaria (Xunta de Galicia, Grant CN2011/002) and FEDER funds.