Extending Topological Abelian Groups by the Unit Circle

and Applied Analysis 3 and any continuous homomorphism t : H → Y, if PO is the push-out of ı and t, there is a commutative diagram


Introduction and Preliminaries
In the theory of topological vector spaces (topological groups) a property is said to be a 3-space property if whenever a closed subspace (subgroup) of a space (group) and the corresponding quotient / both have property , also has property .
A short exact sequence of topological vector spaces (topological groups) 0 → → → → 0 will be called a twisted sum, and the space (group) will be called an extension of by when both and 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 for 0 < < 1, and a weakly closed subspace of without the Hahn-Banach extension property. If we take as the kernel of some continuous linear functional on which does not extend to , then = / does not have the point-separating property but both := / and / have this property (see [1])).
In the category of topological Abelian groups, local compactness, precompactness, metrizability, and completeness are 3-space properties. However, -compactness, sequential completeness, realcompactness, and a number of other properties are not (see [2] for more examples).
The twisted sum 0 → → → → 0 splits if there exists a continuous linear map (continuous homomorphism) : → × making the following diagram commutative ( is the canonical inclusion of into the product, and is the canonical projection onto ): It is known that if such a exists, it must actually be a topological isomorphism.

Abstract and Applied Analysis
If the twisted sum 0 → → → → 0 splits and both and have a productive property , then has property , too. Kalton et al. provided in [1] the first formal and extensive study of splitting twisted sums in the framework of -spaces (complete metric linear spaces). They devote Chapter 5 of this monograph to the following problem: is local convexity a 3space 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 -space by the -space if and only if there exists a non-locally-convex extension of 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 -space is said to be a K-space if, whenever is an -space and is a subspace of with dimension one such that / ≅ , 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 ℓ 1 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 (0 < < ∞), ( ̸ = 1), or 0 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][6][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 the canonical projection from R to T given by ( ) = exp(2 ). We will use N 0 ( ) to denote the system of neighborhoods of 0 in a topological Abelian group .
Recall that a topological Abelian group is precompact if for every neighborhood of zero there exists a finite subset of such that = + . Precompact groups are the subgroups of compact groups. In the same way a group is locally precompact if and only if it is a subgroup of a locally compact group.
The dual group ∧ of a given topological Abelian group is formed from the continuous group homomorphisms from into T, usually called characters. The polar set of a subset of G is defined by 0 := { ∈ ∧ : ( ) ⊆ T + }, where there is a ∈ 0 such that ( ) ∉ 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 ∧ separates the points of we say that 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 ≤ is dually embedded in if each character of can be extended to a character of .
When endowed with the compact-open topology co , ∧ becomes a Hausdorff topological group. A basis of neighborhoods of the neutral element for the compact open topology co is given by the sets 0 = { ∈ ∧ : ( ) ⊆ T + }, where is a compact subset of .

Remark 1.
Observe that a necessary condition for the splitting of the twisted sum of topological Abelian groups 0 → → → → 0 is that ( ) be a dually embedded subgroup of .
The following known characterization is essential when dealing with twisted sums in different categories (see [3,Lemma 3.1] for a proof).
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 , , and and continuous homomorphisms : → and V : → , the push-out of and V is a topological group PO and two continuous homomorphisms and making the square diagram commutative where both squares are commutative and the bottom sequence is a twisted sum [3]. An analogous result for the dual construction (the pullback) can be obtained (see [3]). Proof. Suppose that ∧ separates points of . It is known that for any locally compact Abelian group , a subgroup ≤ ∧ is dense in ∧ if and only if it separates points of (see [9,Proposition 31]). The subgroup of ∧ formed by all restrictions of characters of separates points of by hypothesis. Hence is dense in ∧ and, as ∧ is discrete, coincides with ∧ .
Suppose that is dually embedded in . As is compact, it is a MAP group. Fix a nonzero ∈ . There exists ∈ ∧ such that ( ) ̸ = 1. Since is dually embedded in , there exists an extensioñ∈ ∧ of with̃( ) = ( ) ̸ = 1.

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

The Class TG (T)
Next we consider the particular case in which the compact subgroup 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 → → → 0 be a twisted sum of topological Abelian groups. The following are equivalent: (1) the twisted sum 0 → T → → → 0 splits; (2) ∧ separates points of (T); Proof. 2 ⇔ 3 is a corollary of Lemma 3. 3 ⇒ 1: suppose that (T) is dually embedded in . Hence there exists a continuous character : → T which extends the isomorphism : (T) → T defined by ( ( )) = . Since ∘ = id T , the assertion follows from Theorem 2. 1 ⇒ 2: fix ∈ (T), = ( ) with ̸ = 1. By Theorem 2, there exists a continuous homomorphism : Now we will complete the previous Theorem under the assumption that 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 be a locally quasi-convex subgroup of a topological Abelian group such that / is locally quasi-convex. Then is locally quasi-convex if and only if is dually embedded in .
Theorem 7. Let 0 → T → → → 0 be a twisted sum of topological Abelian groups. Suppose that is locally quasiconvex. Then conditions (1), (2), (3) of Theorem 5 are equivalent to Proof. 1 ⇒ 4: if the twisted sum splits, 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 → → → 0 with and locally quasi-convex, since ≅ / (T), by Lemma 6, we deduce that (T) is dually embedded in .
Following the notation used by Domański [11] in the framework of topological vector spaces, we introduce the class TG (T) which is the analogue of that of K-spaces for topological Abelian groups.
Definition 8. We say that a topological Abelian group is in the class TG (T) if every twisted sum of topological Abelian groups 0 → T → → → 0 splits.

Theorem 9.
Let be a topological vector space such that ∈ TG (T) as a topological group. Then is a K-space.
Proof. Suppose that ∈ TG (T). Fix a twisted sum of topological vector spaces 0 → R → → → 0. Recall that we denote by : R → T the canonical projection, given by ( ) = exp(2 ). If we consider the push-out PO of and , we obtain a commutative diagram where the bottom sequence is a twisted sum. Since ∈ TG (T), this sequence splits. Hence there exists a left inverse for . Then ∘ ∘ = . Since is a topological vector space, ∘ is of the form → exp(2 ( )) for some continuous linear functional ∈ * . This clearly implies ∘ = id R ; that is, is a left inverse for ; 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 → → → 0 which does not split. In other words, ℓ 1 is not a K-space.
Remark 12. The above corollary gives an example of a quotient /T which is locally quasi-convex as a topological group but such that 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 is in TG (T) iff for every twisted sum of topological Abelian groups of the form 0 → T → → → 0, the subgroup (T) is dually embedded in . Note that in any such twisted sum, for an arbitrary ≤ the subgroup −1 ( ) contains (T). This yields the following criterion; Proposition 13. Let be a topological Abelian group. Suppose that there exists a subgroup ≤ which is in TG (T) and satisfies the following property. For every twisted sum of topological Abelian groups of the form 0 → T → → → 0, the subgroup −1 ( ) is dually embedded in . Then ∈ TG (T).

Corollary 14. Let be a topological Abelian group. Suppose that there exists an open subgroup
Proof.
If is an open subgroup of , with the notation of Proposition 13, −1 ( ) is an open subgroup of and hence dually embedded.

Corollary 15.
Let be a topological Abelian group. Suppose that there exists a dense subgroup ≤ such that ∈ TG (T). Then ∈ TG (T).
Proof. If is a dense subgroup of , with the notation of Proposition 13, −1 ( ) is a dense subgroup of because is continuous and onto. In particular it is dually embedded in .
We next see that the converse of Corollary 15 is true in the case of metrizable. For any topological Abelian group , we denote bỹthe Raȋkov completion of . See [14] for more information about this subject. Proof. Let be a topological Abelian group, and let T ≅ ≤ be a subgroup such that ≅ / . Since metrizability is a 3-space property ( [15, 5.38(e)]), is a metrizable group. By Lemma 16,̃=̃≅̃/ . By Corollary 15,̃∈ TG (T). It follows that is dually embedded iñ; hence it is dually embedded in , too.
Our next aim is to prove that Hausdoff locally precompact groups are in TG (T). Note first that if is a topological Abelian group and ≤ is a precompact subgroup such that the quotient / is locally precompact, then is locally precompact, too. (Indeed, let : → / be the canonical projection. Choose ∈ N 0 ( ) such that ( ) is precompact. Let us see that is precompact. Given ∈ N 0 ( ) we need to find a finite subset ⊂ with ⊂ + . Fix ∈ N 0 ( ) with + ⊂ and find a finite 1 ⊂ with ⊂ 1 + . Since ( ) is precompact there exists a finite subset 2 of / with ( ) ⊂ 2 + ( ). We may suppose that 2 = ( 2 ) with 2 being a finite subset of . Hence ( ) ⊂ ( 2 ) + ( ) = Theorem 18. Locally precompact Hausdorff Abelian groups are in TG (T).
Proof. Let be a locally precompact Hausdorff Abelian group. Given a twisted sum 0 → T → → → 0, as T is in particular precompact, by the above argument is locally precompact, too. But every subgroup of a locally compact group is dually embedded [15, 24.12], so (T) is dually embedded iñ, the completion of . Then (T) is also dually embedded in and by Theorem 5, the twisted sum splits.

Corollary 19. Locally compact Hausdorff Abelian groups and precompact Hausdorff Abelian groups are in 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 be a topological Abelian group.
(1) If a closed subgroup ≤ is such that / ∈ TG (T), then is dually embedded.
(2) If is in TG (T) and ≤ is a closed dually embedded subgroup, then / ∈ TG (T). Proof.
(1) Suppose that / ∈ TG (T). Let : → T be a character, and consider the natural twisted sum 0 → → → / → 0. By taking the corresponding push out, we obtain the following commutative diagram: We thus obtain the following diagram: Note that since and are onto, the definition of PB yields (( × ) ∩ PB) ⊃ and (( × ) ∩ PB) ⊃ for every ∈ N 0 ( ) and ∈ N 0 ( ); hence and are onto and open. Thus both short sequences are twisted sums of topological Abelian groups.
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 is in TG (T) if and only if whenever is a topological Abelian group and is a closed subgroup of with / ≅ , then
is dually embedded.
Let ( ) ∈ be a family of topological Abelian groups. The coproduct of ( ) ∈ is the direct sum ⨁ ∈ endowed with the finest group topology making the inclusion maps : is a countable family of groups, this topology coincides with the box topology on ⨁ ∈N . Recall that the coproduct ⨁ ∈ has the following universal property. Given an arbitrary topological Abelian group and a homomorphism : ⨁ ∈ → , is continuous if and only if ∘ is continuous ∀ ∈ .
Proposition 24. Let ( ) ∈ be a family of topological Abelian groups in TG (T). The coproduct ⨁ ∈ is in TG (T).
Proof. Let 0 → T → → ⨁ ∈ → 0 be a twisted sum. Consider, for each ∈ , the pull-back PB of and : → ⨁ ∈ ; for every ∈ there is a commutative diagram  As V ∘ is a continuous homomorphism, by the universal property of the coproduct, is a continuous homomorphism. For every = ( ) ∈ ∈ ⨁ ∈ , we have so is a right inverse for , 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 finitedimensional 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.
(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 TG (T). Then the direct limit lim → is in TG (T). In particular, sequential direct limits of locally compact groups are in TG (T).
with the coproduct topology is in TG (T) by Proposition 24. Let : → ⨁ be the inclusion map, for every ∈ N. It is known (see [17]) that lim → ≅ (⨁ )/ , where is the closure of the subgroup generated by { ∘ ( ) − ( ) : ≤ ; ∈ }. Since countable coproducts of nuclear groups are nuclear groups, is dually embedded. By Theorem 21, lim → ∈ 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][20][21][22][23]).
The following is a relevant fact concerning the structure of L ∞ groups proved by Sulley.

Proposition 26 ([22]). Let be any Abelian group endowed with an L ∞ topology. Then has an open subgroup which is a strict inductive limit of a sequence of Hausdorff locally compact Abelian groups.
Corollary 27. Let be any Abelian group endowed with an L ∞ topology. Then is in TG (T).
Proof. By the above proposition and Theorem 25, has an open subgroup in TG (T). Hence Corollary 14 implies that is in TG (T).

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]).
Our aim is to use the notion of approximable quasihomomorphisms to obtain new examples of groups in TG (T).
The following result is [7,Lemma 11]. We give here a proof for the sake of completeness. Proof. Note that in order to define with ∘ = id , we simply must choose for every ∈ an element ∈ −1 ( ), which is a nonempty set since is onto. Let us see that it can be done in such a way that the map thus obtained is continuous at zero.
Corollary 32. Let be a metrizable topological Abelian group.
(1) If is metrizable and divisible, every twisted sum 0 → → → → 0 is equivalent to one induced by a quasi-homomorphism.
(2) is in TG (T) if and only if every quasi-character : → T is approximable.
Proof. Metrizability is a 3-space property (see [15, 5.38(e)]). If 0 → → → → 0 is a twisted sum where both and are metrizable, so is , and thus the hypotheses of Proposition 31 hold. Therefore, there exists a section of continuous at the origin. As is a divisible group, the twisted sum 0 → → → → 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).
Proof. We can assume, without loss of generality, that is a probability (note that 0 ( ) is topologically isomorphic to 0 (]), where ] is a probability with the same null sets as ).
Proof. This follows from Theorem 39 and Corollary 32, since 0 is a metrizable group.
Example 41. Let 0 be as in Theorem 39. Fix a discrete, nontrivial ≤ 0 (e.g., a copy of Z). Note that 0 does not have any nontrivial continuous character, and in particular is not dually embedded in 0 . Using Theorem 21 we deduce that 0 / is not in TG (T). Since 0 ∈ TG (T), this example shows that being in TG (T) is not preserved by local isomorphisms (compare with Corollary 14).
Let ( , ) be a topological group. We say that ( , ) is a protodiscrete group (or that the topology 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 be a protodiscrete topological Abelian group. Every quasi-character of is approximable.
Proof. Let : → T be a quasi-character. There exists an open subgroup ≤ such that ( + ) ( ) ( ) ∈ T + for every , ∈ . Using Lemma 38 we deduce that there exists an algebraic character : → T with ( ) ( ) ∈ T + for every ∈ . Now Corollary 37 implies that any algebraic extension of approximates .

Corollary 43. Every protodiscrete, metrizable group is in
Proof. This follows from Proposition 42 and Corollary 32.
Example 44. Countable products of discrete Abelian groups belong to TG (T).