Topological groups with invariant linear spans
aa r X i v : . [ m a t h . GN ] J u l TOPOLOGICAL GROUPS WITH INVARIANT LINEAR SPANS
EVA PERNECK ´A AND JAN SPˇEV ´AK
Abstract.
Given a topological group G that can be embedded as a topological subgroup intosome topological vector space (over the field of reals) we say that G has invariant linear span ifall linear spans of G under arbitrary embeddings into topological vector spaces are isomorphicas topological vector spaces.For an arbitrary set A let Z ( A ) be the direct sum of | A | -many copies of the discrete groupof integers endowed with the Tychonoff product topology. We show that the topological group Z ( A ) has invariant linear span. This answers a question from [1] in positive.We prove that given a non-discrete sequential space X , the free abelian topological group A ( X ) over X is an example of a topological group that embeds into a topological vector spacebut does not have invariant linear span. All vector spaces in this paper are considered over the field R of real numbers and all topo-logical spaces are assumed to be Hausdorff. For an arbitrary non-empty set A and a topologicalgroup G with addition and neutral element 0 G let G A be the topological group given by thedirect product Π a ∈ A G with coordinate-wise addition and the Tychonoff product topology. Wedenote G ( A ) the topological subgroup of G A , with inherited topology, consisting of those ele-ments ( g a ) a ∈ A for which the set { a ∈ A : g a = 0 G } is finite. Given a subset H of a group G anda subset M of a vector space L , we use the standard notation h H i to denote the subgroup of G generated by H and span( M ) for the vector subspace of L generated by M . For simplicity wewrite h g i rather than h{ g }i for any g ∈ G and, similarly, span( l ) instead of span( { l } ) for any l ∈ L . 1. Introduction
In this note we study which topological groups enjoy the property stated in the followingdefinition.
Definition 1.1.
Let G be a topological group that can be embedded (as a topological subgroup)into some topological vector space. We say that G has invariant linear span provided that alllinear spans of G under arbitrary embeddings into topological vector spaces are isomorphic astopological vector spaces.A simple example of topological group with an invariant linear span is every topological vectorspace. Indeed, as was observed by Tkachuk in [9], given arbitrary topological vector spaces L and E and a continuous group homomorphism h : L → E , the homomorphism h is automaticallylinear. This observation further yields that if L is embedded in E as a topological subgroup, thesame embedding is already an embedding of topological vector spaces. In particular, the linearspan of L in E is (isomorphic to) the topological vector space L again and hence the linearspan of L does not depend on the space E in which L embeds. Yet another simple example ofa topological group with an invariant linear span is the discrete topological group Z of integers.Its linear span is obviously (isomorphic to) the topological vector space R .In our paper we show that for an arbitrary non-empty set A the group Z ( A ) has invariantlinear span (which is isomorphic to R ( A ) ). See Theorem 3.4 and Corollary 3.5. This answers [1,Question 10.6] in positive and generalizes the folklore fact that all topological vector spaces ofthe same finite dimension are isomorphic (see Remark 3.6).The proof of Theorem 3.4 consists of two steps. The first was done in [1, Proposition 10.1]by showing that given an injective linear map l : R ( A ) → L , where L is a topological vector The first author was supported by the grant GA ˇCR 18-00960Y of the Czech Science Foundation. space, the continuity of l follows from the continuity of the restriction of l to Z ( A ) . The secondstep is done in Theorem 2.4, where we basically show, that if the restriction of l to Z ( A ) isan embedding of topological groups, then l is open. The proof of Theorem 2.4 is based ona Diophantine approximation done in Lemma 2.2 which resembles the classical Kronecker’sapproximation theorem.We end the paper with Theorem 3.7, which shows that for an arbitrary non-discrete sequentialspace X the free topological abelian group A ( X ) does not have invariant linear span, as itcanonically embeds in both the free topological vector space V ( X ) and the free locally convextopological vector space L ( X ), and the linear spans of A ( X ) in the latter spaces are the non-isomorphic topological vector spaces V ( X ) and L ( X ). This theorem is based on non-trivialresults of Tkachenko [8] and Gabriyelyan and Morris [2].2. The main technical theorem
We begin the section by two auxiliary observations.
Lemma 2.1.
For every neighbourhood V of zero in a compact group G and every t ∈ G thereis m ∈ N \ { } such that (1) mt ∈ V. Proof.
Pick a neighbourhood V of zero in G and t ∈ G arbitrarily. There are two possibilities.If h t i is a discrete subgroup of G then it is closed and, consequently, compact and thereforefinite. Let m be the order of h t i and observe that (1) holds. The second possibility is, that h t i isnot discrete. Then every neighbourhood of zero (and V in particular) contains infinitely manyelements of h t i . Since t is a generator of h t i there is m ∈ N \ { } satisfying (1). (cid:3) Lemma 2.2.
Let ( t n ) n ∈ N be a sequence in R F , where F is a finite set. For every neighborhood O of zero in R F there is m ∈ N \ { } and a sequence ( z n ) n ∈ N ⊂ Z F such that (2) mt n − z n ∈ O holds for infinitely many n ∈ N .Proof. Fix O , a neighbourhood of zero in R F , arbitrarily, and let q : R F → ( R / Z ) F be thequotient map. Since ( R / Z ) F is sequentially compact, the sequence ( q ( t n )) n ∈ N has a convergentsubsequence with a limit t . As q is an open map, we may pick a neighbourhood V of zero in( R / Z ) F such that V + V ⊂ q ( O ) . By Lemma 2.1, there is a positive integer m satisfying (1). Observe that the set M defined as M := { n ∈ N : mq ( t n ) ∈ mt + V } is infinite, and for every n ∈ M we have q ( mt n ) = mq ( t n ) ∈ mt + V ⊂ V + V ⊂ q ( O ) . Thus for every n ∈ M there is z n ∈ Z F such that (2) holds. (cid:3) In order to formulate the main technical result of this paper we need to recall three notions.Their importance to the topic of our manuscript will become clear from Proposition 3.3 andfrom the proof of Theorem 3.4.We say that a subset A of a topological vector space L is • absolutely Cauchy summable provided that for every neighbourhood V of 0 L there existsa finite set F ⊂ A such that(3) span( A \ F ) ⊂ V ; • topologically independent if 0 L A and for every neighbourhood W of 0 L there exists aneighbourhood U of 0 L such that for every finite subset F ⊂ A and every indexed set { z a : a ∈ F } of integers the inclusion P a ∈ F z a a ∈ U implies that z a a ∈ W for all a ∈ F .We call this neighbourhood U a W -witness of the topological independence of A ; OPOLOGICAL GROUPS WITH INVARIANT LINEAR SPANS 3 • semi-basic if for all a ∈ A we have(4) a span( A \ { a } ) . Remark 2.3.
In [1, Definition 3.1] the notion of an absolutely Cauchy summable set wasintroduced in an arbitrary abelian topological group. In topological vector spaces it is equivalentto our definition by [1, Proposition 9.2 (i)].
Topologically independent sets were introduced in [1, Definition 4.1] in an arbitrary abeliantopological group. For further properties of these sets in precompact groups we refer to [7].We have adopted the name semi-basic from [5], where a semi-basic sequence in an F -spacewas introduced. In [4] a semi-basic set is called topologically free . Semi-basic sequences inBanach spaces are called minimal in [3] and [6, Definition 6.1].Now we are ready to state the main technical theorem of this note. Theorem 2.4. If A is a topologically independent and absolutely Cauchy summable subset ofa topological vector space L , then A is semi-basic.Proof. To prove the contrapositive, assume that there is a ∈ A with(5) a ∈ span( A \ { a } ) , and let A be absolutely Cauchy summable. We will show that A is not topologically independent.If a = 0 L , then we are done. Otherwise we can find a neighbourhood W of 0 L such that za / ∈ W for every z ∈ Z \ { } . Pick an arbitrary neighborhood U of 0 L . Let us show that U isnot a W -witness of topological independence of A .Fix a balanced neighborhood V of 0 L with V + V + V ⊂ U . Since A is absolutely Cauchysummable, there is a finite F ⊂ A \ { a } such that span( A \ ( F ∪ { a } )) ⊂ V. In particular, forevery finite B ⊂ A \ { a } , reals ( s b ) b ∈ B \ F and each n ∈ N we have(6) X b ∈ B \ F s b b ∈ n V. Given n ∈ N arbitrarily, by (5) we can fix a finite set B ⊂ A \ { a } and an indexed set ( r nb ) b ∈ B of reals such that(7) a − X b ∈ B r nb b ∈ n V. For b ∈ F \ B define r nb = 0, and observe that (6) and (7) yield(8) a − X b ∈ F r nb b = a − X b ∈ B r nb b ! + X b ∈ B \ F r nb b ∈ n V + 1 n V. By continuity of vector space operations, there is a neighborhood O of zero in R F such that(9) X b ∈ F s b b ∈ V for all ( s b ) b ∈ F ∈ O. Define a sequence ( t n ) n ∈ N in R F by t n = ( r nb ) b ∈ F , and let m ∈ N \ { } and ( z n ) n ∈ N ⊂ Z F be asin the conclusion of Lemma 2.2. By this lemma, we may fix n ∈ N such that n ≥ m and (2)holds. For b ∈ F let z b ∈ Z be the b -th coordinate of z n , and observe that by (2) and (9) wehave X b ∈ F ( mr nb − z b ) b ∈ V. From this, (8), and the fact that V is balanced and n ≥ m we get ma − X b ∈ F z b b = m a − X b ∈ F r nb b ! + X b ∈ F ( mr nb − z b ) b ! ∈ mn V + mn V + V ⊂ V + V + V ⊂ U. Since m is a non-zero integer and z b is an integer for each b ∈ F we conclude that U is not a W -witness of the topological independence of A , because ma W by the choice of W . (cid:3) E. PERNECK ´A AND J. SPˇEV ´AK The invariance of the linear span of Z ( A ) In this section we prove that the topological group Z ( A ) has invariant linear span. In orderto do so we need to recall the notion of a (linear) Kalton map introduced in [1] which is usefulto deal with embeddings of Z ( A ) and R ( A ) into topological vector spaces.Given a non-empty subset A of a topological vector space L such that 0 A , we denote K A : Z ( A ) → L the group homomorphism given by K A (( z a ) a ∈ A ) = P a ∈ A z a a for every ( z a ) a ∈ A ∈ Z ( A ) . Simi-larly, ℓ K A : R ( A ) → L is the linear operator between vector spaces defined by ℓ K A (( r a ) a ∈ A ) = P a ∈ A r a a for every( r a ) a ∈ A ∈ R ( A ) . As in [1] we call K A ( ℓ K A ) the (linear) Kalton map associated with A . Sincethe sums in the definitions are finite, the mappings are well-defined and K A ( Z ( A ) ) = h A i ⊂ L and ℓ K A ( R ( A ) ) = span( A ) ⊂ L . Notice that the (linear) Kalton map is injective if and only if A is (linearly) independent. Fact 3.1 ([1, Proposition 10.1]) . Given a non-empty subset A of non-zero elements of a topo-logical vector space, the following statements are equivalent: (i) the linear Kalton map ℓ K A is continuous; (ii) the Kalton map K A is continuous; (iii) the set A is absolutely Cauchy summable. Lemma 3.2.
Let A be a non-empty subset of a topological vector space. The following conditionsare equivalent: (i) the linear Kalton map ℓ K A is an open injection onto span( A ) ; (ii) the set A is semi-basic.Proof. Observe that from both items (i) and (ii) it follows that A is linearly independent.Therefore, if we assume either (i) or (ii), then for each a ∈ A there is a unique linear projection π Aa : span( A ) → span( a ) such that ker π Aa = span( A \ { a } ) and π Aa restricted to span( a ) is theidentity map.To end the proof it suffices to show that items (i) and (ii) are both equivalent to the followingfact for a linearly independent set A :(10) π Aa : span( A ) → span( a ) is continuous for every a ∈ A .The equivalence of (i) and (10) follows from [1, Proposition 10.2]. To establish the equivalenceof (ii) and (10) it suffices to realize that the continuity of each π Aa is equivalent to the fact thateach ker( π Aa ) is closed in span( A ) and this happens if and only if (4) holds for all a ∈ A . (cid:3) Proposition 3.3.
Given a subset A of a topological vector space, the following conditions areequivalent: (i) The linear Kalton map ℓ K A is an embedding of topological vector spaces. (ii) A is absolutely Cauchy summable and semi-basic.Proof. Assume (i). Then A is absolutely Cauchy summable by Fact 3.1 and semi-basic byLemma 3.2. Thus (ii) holds.If (ii) holds, Lemma 3.2 implies, that the linear Kalton map ℓ K A is open and injective, whileFact 3.1 provides its continuity. This gives us (i). (cid:3) Our next theorem answers [1, Question 10.6] in positive.
Theorem 3.4.
Given a subset A of a topological vector space the following statements areequivalent: (i) The Kalton map K A is an embedding of topological groups; (ii) The linear Kalton map ℓ K A is an embedding of topological vector spaces. OPOLOGICAL GROUPS WITH INVARIANT LINEAR SPANS 5
Proof.
Since K A is a restriction of ℓ K A , the implication (ii) ⇒ (i) follows.Assume (i). Then A is absolutely Cauchy summable by Fact 3.1. Further, A is topologicallyindependent by [1, Proposition 4.7 (ii)]. Theorem 2.4 yields that A is also semi-basic. To show(ii) it remains to apply Proposition 3.3. (cid:3) The next statement is a direct corollary of Theorem 3.4.
Corollary 3.5.
For every non-empty set A the topological group Z ( A ) has invariant linear span(which is isomorphic to R ( A ) ). Remark 3.6.
Corollary 3.5 can be viewed as a generalization of the folklore fact that alltopological vector spaces of the same finite dimension are isomorphic . Indeed, if A is a finite basisof a topological vector space V , then A is topologically independent by [1, Proposition 4.11].It follows then by [1, Proposition 4.8] that the Kalton map K A is an embedding of topologicalgroups. That is, the hull h A i is (isomorphic to) Z A . Hence V = span( A ) is (isomorphic to) R A .We end this paper with a theorem which provides a rich source of examples of topologicalgroups that embed in topological vector spaces and do not have invariant linear spans.Given a Tychonoff space X the symbols A ( X ), L ( X ) and V ( X ) stand for the free abeliantopological group, the free locally convex topological vector space and the free topological vectorspace over X respectively. We refer the reader to [2] for definitions of these notions. Theorem 3.7.
Let X be a Tychonoff space. The topological group A ( X ) canonically embeds inthe topological vector spaces L ( X ) and V ( X ) . If X is sequential and non-discrete, then A ( X ) does not have an invariant linear span.Proof. By [8, Theorem 3] the topological group A ( X ) embeds in L ( X ) and the linear span of A ( X ) is L ( X ). On the other hand, by [2, Proposition 5.1], it also embeds in V ( X ) and its linearspan in V ( X ) is V ( X ). Finally, if L ( X ) and V ( X ) are isomorphic as topological vector spacesand X is sequential, then X is discrete by [2, Corollary 4.5]. (cid:3) References [1] D. Dikranjan, D. Shakhmatov and J. Spˇev´ak,
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E-mail address : [email protected] (Jan Spˇev´ak) Department of Applied Mathematics, Faculty of Information Technology, CzechTechnical University in Prague, Th´akurova 9, 16000, Prague 6, Czech Republic
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