#### Abstract

A twisted sum in the category of topological Abelian groups is
a short exact sequence where all maps are assumed to be continuous and open onto their images. The twisted sum splits if it is equivalent to . We study the class of topological groups *G* for which every twisted sum splits. We prove that this class contains Hausdorff locally precompact groups, sequential direct limits of locally compact groups, and topological groups with 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 , we use the connection between extensions of the form 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 -space, which were interpreted for topological groups by Cabello.

#### 1. 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) 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 , 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 * 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 ):
(1)

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

If the twisted sum 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 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 -space by the -space if and only if there exists a non-locally-convex extension of by . 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 -*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 is not a -space.

The notion of -space is relevant on its own, regardless of 3-space properties. Many classical spaces such as (), (), or are -spaces.

In this paper we will study the natural counterpart of the notion of -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 the neutral element. We denote by the set of all positive natural numbers, by the integers, by the reals, by the set of complex numbers, and by the unit circle of , with the topology induced by . In we will use multiplicative notation and we will denote by the canonical projection from to given by . We will use to denote the system of neighborhoods of 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 , usually called *characters*. The polar set of a subset of is defined by , where . A subset of a topological group is called *quasi-convex* if for every there is a such that . A topological group is called *locally quasi-convex* if it has a neighborhood basis of 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 , becomes a Hausdorff topological group. A basis of neighborhoods of the neutral element for the compact open topology is given by the sets , where is a compact subset of .

*Remark 1. *Observe that a necessary condition for the splitting of the twisted sum of topological Abelian groups 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).

Theorem 2. *Let be a twisted sum of topological Abelian groups. The following are equivalent: *(1)*;
*(2)* there exists a continuous homomorphism with , i.e., a right inverse for ;*(3)

*there exists a continuous homomorphism*with ,*i.e., a left inverse for*.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 , the push-out of and is a topological group and two continuous homomorphisms and making the square diagram commutative (2)

and such that for every topological Abelian group and continuous homomorphisms and with , there is a unique continuous homomorphism from to making the two triangles commutative. The topological group exits and is unique up to topological isomorphism.

Given any twisted sum of topological Abelian groups, any topological Abelian group , and any continuous homomorphism , if is the push-out of and , 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 be a compact subgroup of a topological Abelian group . separates points of if and only if is dually embedded in .*

*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 . Since is dually embedded in , there exists an extension of with .

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 .*

#### 2. The Class

Next we consider the particular case in which the compact subgroup is . (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 be a twisted sum of topological Abelian groups. The following are equivalent: *(1)*the twisted sum splits; *(2)* separates points of ; *(3)* is dually embedded in . *

*Proof. * is a corollary of Lemma 3.

: suppose that is dually embedded in . Hence there exists a continuous character which extends the isomorphism defined by . Since , the assertion follows from Theorem 2.

: fix , with . By Theorem 2, there exists a continuous homomorphism with ; hence .

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 be a twisted sum of topological Abelian groups. Suppose that is locally quasi-convex. Then conditions (1), (2), (3) of Theorem 5 are equivalent to *(4)* is locally quasi-convex.*

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

: given a twisted sum with and locally quasi-convex, since , by Lemma 6, we deduce that is dually embedded in .

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

*Definition 8. *We say that a topological Abelian group is in the class if every twisted sum of topological Abelian groups splits.

Theorem 9. *Let be a topological vector space such that as a topological group. Then is a -space.*

*Proof. *Suppose that . Fix a twisted sum of topological vector spaces . Recall that we denote by the canonical projection, given by . If we consider the push-out of and , we obtain a commutative diagram
(4)

where the bottom sequence is a twisted sum. Since , this sequence splits. Hence there exists a left inverse for . Then . Since is a topological vector space, is of the form for some continuous linear functional . This clearly implies ; 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 which does not split. In other words, is not a -space.*

Using Theorem 9 we deduce the following.

Corollary 11. *.*

*Remark 12. *The above corollary gives an example of a quotient which is locally quasi-convex as a topological group but such that does not even separate points of : 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 iff for every twisted sum of topological Abelian groups of the form , the subgroup is dually embedded in . Note that in any such twisted sum, for an arbitrary the subgroup contains . This yields the following criterion;

Proposition 13. *Let be a topological Abelian group. Suppose that there exists a subgroup which is in and satisfies the following property. For every twisted sum of topological Abelian groups of the form , the subgroup is dually embedded in . Then .*

Corollary 14. *Let be a topological Abelian group. Suppose that there exists an open subgroup such that . Then .*

*Proof. *If is an open subgroup of , with the notation of Proposition 13, 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 . Then .*

*Proof. *If is a dense subgroup of , with the notation of Proposition 13, 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 by the Raĭkov completion of . See [14] for more information about this subject.

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

Proposition 17. *Let be a metrizable topological Abelian group which is in . Suppose that is a dense subgroup of . Then is in , too.*

*Proof. *Let be a topological Abelian group, and let 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, . It follows that is dually embedded in ; hence it is dually embedded in , too.

Our next aim is to prove that Hausdoff locally precompact groups are in . 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 such that is precompact. Let us see that is precompact. Given we need to find a finite subset with . Fix with and find a finite with . Since is precompact there exists a finite subset of with . We may suppose that with being a finite subset of . Hence , which implies if we put .)

Theorem 18. *Locally precompact Hausdorff Abelian groups are in .*

*Proof. *Let be a locally precompact Hausdorff Abelian group. Given a twisted sum , as 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 is dually embedded in , the completion of . Then 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 . *

*Remark 20. *It is proved in [11] that every topological vector space endowed with its weak topology is a -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 , then is dually embedded. *(2)*If is in and is a closed dually embedded subgroup, then . *

*Proof. *(1) Suppose that . Let be a character, and consider the natural twisted sum . By taking the corresponding push out, we obtain the following commutative diagram:(5)

where both rows are twisted sums of topological Abelian groups. Since , the bottom sequence splits; hence by Theorem 2, there exists a continuous homomorphism such that . The homomorphism is an extension of .

(2) Suppose that is dually embedded in . Fix a twisted sum . Let be the canonical projection, and let be the pull-back of and . Define and , as the restrictions of the corresponding projections. Note that

We thus obtain the following diagram:
(7)

Note that since and are onto, the definition of yields and for every and ; hence and are onto and open. Thus both short sequences are twisted sums of topological Abelian groups.

Fix ; we will find an extension of to the whole . We can regard as a character of by defining . Since and , by Theorem 5, is dually embedded in . Thus there exists with ; that is, for every . Define, for every , (note that if then ).

Since is dually embedded in there exists with i.e. for every . Now define as follows: . This is clearly continuous. Note that since if , we have that . As , the character in given by for every is well defined and continuous.

Now is the desired extension of : if , we have .

*Remark 22. *An analogous result in the framework of -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 is in if and only if whenever is a topological Abelian group and is a closed subgroup of with , then is dually embedded.*

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

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 continuous, for every . If is a countable family of groups, this topology coincides with the box topology on .

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 . The coproduct is in .*

*Proof. *Let be a twisted sum. Consider, for each , the pull-back of and ; for every there is a commutative diagram
(8)whose rows are twisted sums. As , by Theorem 2, there exists a homomorphism such that . Consider the map
As 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 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 . Then the direct limit is in . In particular, sequential direct limits of locally compact groups are in .*

*Proof. *The direct sum with the coproduct topology is in by Proposition 24. Let be the inclusion map, for every . It is known (see [17]) that , where is the closure of the subgroup generated by . Since countable coproducts of nuclear groups are nuclear groups, is dually embedded. By Theorem 21, .

Varopoulos introduced in [18] the class 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 groups. Subsequently many other authors investigated different properties of this class ([19–23]).

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

Proposition 26 ([22]). *Let be any Abelian group endowed with an 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 topology. Then is in .*

*Proof. *By the above proposition and Theorem 25, has an open subgroup in . Hence Corollary 14 implies that is in .

#### 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 and be topological Abelian groups and a map with . We say that is a * quasi-homomorphism* if the map is continuous at .

A quasi-homomorphism is * approximable* if there exists a homomorphism such that is continuous at .

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

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

Proposition 29. *Let and be topological Abelian groups and a quasi-homomorphism. *(1) The sets
form a basis of neighborhoods of zero for a group topology on .(2) If denotes the group endowed with the topology induced by the quasi-homomorphism and and 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 such that , and is continuous at the origin.

Lemma 30. *Let and be topological Abelian groups and a quasi-homomorphism. Then the map is continuous at .*

*Proof. *Note that for every and we have .

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

Proposition 31. *Let be a continuous and open surjective homomorphism between topological Abelian groups. Suppose that is metrizable. Then there exists such that and is continuous at .*

*Proof. *Note that in order to define with , we simply must choose for every an element , 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.

Let be a decreasing basic sequence of neighborhoods of zero in , where . Due to the continuity of , we have . Let take the value on . For any , by the previous paragraph, we can choose and with , and define . Now fix ; we must find with . Since is open there exists with . Fix , and let us show that . If this is trivial. Otherwise with for some . Then ; hence and .

Corollary 32. *Let be a metrizable topological Abelian group. *(1)* If is metrizable and divisible, every twisted sum is equivalent to one induced by a quasi-homomorphism. *(2)* is in if and only if every quasi-character is approximable. *

*Proof. *Metrizability is a 3-space property (see [15, 5.38(e)]). If 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 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 any subadditive, symmetric functional such that if and only if .

*Definition 33. *For any let us call the only real number such that . Then is a norm on . (Note that ; hence .) Put . For every let us define . Note that with this notation.

*Definition 34. *For every Abelian group , every with , and every we define .

Proposition 35 ([24, Corollary 2]). *For every and , we have .*

Lemma 36. *Let be a topological Abelian group, and let be a quasi-character. Suppose that there exist and such that . Then is continuous at zero.*

*Proof. *Since is a quasi-character, for every , there exists with for every . Fix any . Let us find with .

Fix any . Find with . Put .

It is enough to prove that

since by Proposition 35, this will imply . Now, for every
Since , we have that . Now, for every and every , , since . Thus , and we deduce . This implies that .

Corollary 37. *A quasi-character is approximable if and only if there exist , and an algebraic character such that . Moreover any such approximates .*

Lemma 38 ([7, Lemma 6]). *Let be an Abelian group (no topology is assumed), and let be any mapping such that for some , . Then there is a unique character such that .*

Theorem 39. *Let be a nonatomic -finite measure on a set . Let be the space of all measurable functions on with the norm
**
Then every quasi-character is approximable.*

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

Let be a quasi-character, fix , and choose such that for every with .

Let be a partition of into measurable sets, with for all . Then as a topological direct product. For all we have that
Call the restriction of to each . As for every and , we can apply Lemma 38 to obtain unique characters such that for every . By Lemma 36, we have that each is continuous at the origin of , and, thus, the character given by (where ) approximates near the origin.

Corollary 40. * is in .*

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

*Example 41. *Let be as in Theorem 39. Fix a discrete, nontrivial (e.g., a copy of ). Note that does not have any nontrivial continuous character, and in particular is not dually embedded in . Using Theorem 21 we deduce that is not in . Since , this example shows that being in 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 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 be a quasi-character. There exists an open subgroup such that for every . Using Lemma 38 we deduce that there exists an algebraic character with 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 .

#### 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.