PONTRYAGIN DUALITY FOR TOPOLOGICAL ABELIAN GROUPS 1

PONTRYAGIN DUALITY FOR TOPOLOGICAL ABELIAN GROUPS ´ SALVADOR HERNANDEZ

Abstract. A topological abelian group 𝐺 is Pontryagin reflexive, or P-reflexive for short, if the natural homomorphism of 𝐺 to its bidual group is a topological isomorphism. In this paper we look at the question, set by Kaplan in 1948, of characterizing the topological Abelian groups that are P-reflexive. Thus, we find some conditions on an arbitrary group 𝐺 that are equivalent to the P-reflexivity of 𝐺 and give an example that corrects a wrong statement appearing in previously existent characterizations of P-reflexive groups. Some other related properties of topological Abelian groups are also considered.

1. introduction In this paper we look at the question, set by Kaplan in 1948, of characterizing the topological Abelian groups for which Pontryagin duality holds. Some other related properties of topological groups are also considered. To begin with we shall describe briefly the elements of the theory. Let (𝐺, 𝜏 ) be an arbitrary topological abelian group. A character on (𝐺, 𝜏 ) is a continuous function 𝜒 from 𝐺 to the complex numbers of modulus one 𝕋 such that 𝜒(𝑥𝑦) = 𝜒(𝑥)𝜒(𝑦) for all 𝑥 and 𝑦 in 𝐺. The pointwise product of two characters ˆ of all characters is a group with pointwise is again a character, and the set 𝐺 ˆ is equipped with the compact open multiplication as the composition law. If 𝐺 ˆ topology, it becomes a topological group (𝐺, 𝜏ˆ) which is called the dual group of (𝐺, 𝜏 ). There is a natural evaluation homomorphism (not necessarily continuous) ˆˆ 𝑒𝐺 : 𝐺 −→ 𝐺 of 𝐺 to its bidual group. A topological abelian group (𝐺, 𝜏 ) satisfies Pontryagin duality or, equivalently, is Pontryagin reflexive (P-reflexive for short,) if the evaluation map 𝑒𝐺 is a topological isomorphism onto. If 𝑒𝐺 is just bijective, we say that 𝐺 is P-semireflexive. The Pontryagin-van Kampen theorems says that every locally compact abelian group satisfies Pontryagin duality. The notion of duality, in one form or another, has always been a crucial point in mathematics and Pontryagin duality is one best example of that; (for the significance of Pontryagin duality for the rest of mathematics, one should consult Mackey [12].) This important result has been subsequently extended to more general classes of topological groups. Kaplan [8, 9] proved that the class of P-reflexive groups is closed under arbitrary products and direct sums and he set the problem of characterizing the class of topological abelian groups for which the Pontryagin duality holds. Smith [15] proved that the additive groups 1991 Mathematics Subject Classification. Primary 05C38, 15A15; Secondary 05A15, 15A18. Key words and phrases. Keyword one, keyword two, keyword three. Research partially supported by Spanish DGES, grant number PB96-1075, and Fundaci´ o Caixa Castell´ o, grant number P1B98-24. 1

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of Banach spaces and reflexive locally convex vector spaces are P-reflexive. More recently, Banaszczyk [2] has considered this question within the class of nuclear groups. Pontryagin duality for free topological abelian groups and spaces of continuous functions has been investigated in [4, 6]. Motivated by Kaplan’s question, Venkatamaran [19] found a characterization of the groups satisfying Pontryagin duality, however, this characterization is very technical and, moreover, contains a wrong statement. A nicer characterization of P-reflexive groups was found by Kye [11] for the class of additive groups of locally convex spaces but again the characterization is incomplete since it contains a wrong statement which is analogous to the one in Venkatamaran’s result. In both cases it is the P-semireflexivity of the groups concerned what is really characterized. But, even in this case, a simpler characterization is desirable for general topological abelian groups. Recent contributions related to the characterization of 𝑃 -reflexive groups are [1, 4, 6, 16]. In the present paper we show a characterization of the topological abelian groups that satisfy Pontryagin duality and we give an example that corrects the wrong statement that appears in the characterizations given by Venkatamaran [19] and Kye[11]. Our main tool has been the notion of ”groups in duality”. This is a crucial concept of Functional Analysis that was first introduced by Varopoulos (cf. [17]) in the context of topological abelian groups. 2. groups in duality Definition 1. (Varopoulos) Let 𝐺 and 𝐺′ be two abelian groups then we say that they are in duality if and only if there is a function ⟨., .⟩

:

′

𝐺 × 𝐺′ −→ 𝕋 ′

such that ′

⟨𝑔1 𝑔2 , 𝑔 ⟩ =

⟨𝑔1 , 𝑔 ⟩ ⋅ ⟨𝑔2 , 𝑔 ⟩

⟨𝑔, 𝑔1′ 𝑔2′ ⟩ =

⟨𝑔, 𝑔1′ ⟩ ⋅ ⟨𝑔, 𝑔2′ ⟩

for all 𝑔1 , 𝑔2 , 𝑔 ∈ 𝐺 and 𝑔 ′ , 𝑔1′ 𝑔2′ ∈ 𝐺′ and it holds: (i) if 𝑔 ∕= 0𝐺 , the neutral element of 𝐺, then there exists 𝑔 ′ ∈ 𝐺′ such that ⟨𝑔, 𝑔 ′ ⟩ = ∕ 1; and (ii) if 𝑔 ′ = ∕ 0𝐺′ , the neutral element of 𝐺′ , there exists 𝑔 ∈ 𝐺 such that ⟨𝑔, 𝑔 ′ ⟩ = ∕ 1. Definition 2. (Varopoulos) If we have a duality ⟨𝐺, 𝐺′ ⟩ we say that a topology 𝜏 on 𝐺 is compatible with the duality when (𝐺, 𝜏 )ˆ = 𝐺′ . Thus we can say that a topological abelian group (𝐺, 𝜏 ) is maximally almost ˆ which periodic (MAP) when the topology 𝜏 is compatible with the duality ⟨𝐺, 𝐺⟩ ˆ is defined by ⟨𝑔, 𝜒⟩ = 𝜒(𝑔) for all 𝑔 ∈ 𝐺 and 𝜒 ∈ 𝐺. Notice that every P-reflexive group must be maximally almost periodic. For a given duality 𝛼 = ⟨𝐺, 𝐺′ ⟩ we may associate two canonical topologies on 𝐺 and 𝐺′ respectively. The topology 𝑤(𝐺, 𝐺′ ) on 𝐺 is the weak topology generated by all the elements in 𝐺′ considered as continuous homomorphisms into 𝕋. The topology 𝑤(𝐺′ , 𝐺) is defined similarly. Both topologies are totally bounded and compatible with the given duality, therefore, their completions are compact groups denoted by 𝑏(𝐺, 𝐺′ ) and 𝑏(𝐺′ , 𝐺) respectively (cf. [17].) For any subset 𝐴 in 𝐺, we denote by 𝐴𝛼 to the set of all 𝑔 ′ ∈ 𝐺′ such that Re⟨𝑔, 𝑔 ′ ⟩ ≥ 0 for all 𝑔 ∈ 𝐴, where Re𝑧 means the real part of the complex number 𝑧. Analogously, if 𝐵 ′ ⊂ 𝐺′ , then 𝐵𝛼′ is the set of all 𝑔 ∈ 𝐺 such that Re⟨𝑔, 𝑔 ′ ⟩ ≥ 0 for all 𝑔 ′ ∈ 𝐵 ′ . These two operators behave in many aspects like the polar operator in vector spaces. For instance, it is easily checked that ((𝐴𝛼 )𝛼 )𝛼 = 𝐴𝛼 for any 𝐴 ⊂ 𝐺

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and ((𝐵𝛼′ )𝛼 )𝛼 = 𝐵𝛼′ for any 𝐵 ′ ⊂ 𝐺′ . Given an arbitrary subset 𝐴 in 𝐺, we define the 𝛼-convex hull of 𝐴, denoted 𝑐𝑜𝛼 (𝐴), as the set (𝐴𝛼 )𝛼 . A set 𝐴 is said 𝛼-convex when it coincides with its 𝛼-convex hull. It is easy to see that different dualities give place to different ”convex” hulls; for example, if 𝛼 = ⟨ℚ, ℝ⟩ and 𝛽 = ⟨ℝ, ℝ⟩ and both are equipped with the standard bilinear mapping defined by ⟨𝑎, 𝑏⟩ = 𝑎𝑏 then, defining 𝐿 = [−1, 1] ∩ ℚ, it holds that 𝑐𝑜𝛼 (𝐿) = 𝐿 but 𝑐𝑜𝛽 (𝐿) = [−1, 1]. Given a duality 𝛼 = ⟨𝐺, 𝐺′ ⟩, if 𝜏 is a topology on 𝐺, we say that 𝜏 is locally 𝛼-convex when there is a neighborhood base of the identity consisting of 𝛼-convex sets. It is readily verified that 𝑤(𝐺, 𝐺′ ) is the weakest locally 𝛼-convex topology compatible with the duality 𝛼. In [17] Varopoulos found the following characterization of a compatible topology. Theorem 1. (Varopoulos) Let 𝛼 = ⟨𝐺, 𝐺′ ⟩ be a duality and let us denoted by 𝛽 the duality ⟨𝐺, 𝑏(𝐺′ , 𝐺)⟩. Then a necessary and sufficient condition for an arbitrary topology 𝜏 on 𝐺 to be compatible with the duality 𝛼 is that, for every neighborhood 𝑈 of the identity in 𝜏 , it holds 𝑈 𝛽 = 𝑈 𝛼 and 𝜏 ≥ 𝑤(𝐺, 𝐺′ ). As a consequence of this result it follows the following simple characterization of continuity. Corollary 1. With the same hypothesis as in the Theorem above, it holds that a necessary and sufficient condition for an arbitrary element 𝜒 ∈ 𝑏(𝐺′ , 𝐺) to be in 𝐺′ is that {𝜒}𝛽 is a neighborhood of the identity in some topology 𝜏 on 𝐺 compatible with 𝛼. Proof. If 𝑈 := {𝜒}𝛽 is a neighborhood of the identity in some topology 𝜏 on 𝐺 compatible with 𝛼, then 𝜒 ∈ 𝑈 𝛽 = 𝑈 𝛼 ⊂ 𝐺′ . Conversely, if 𝜒 ∈ 𝐺′ , then {𝜒}𝛽 = {𝜒}𝛼 is a neighborhood of the identity in the topology 𝑤(𝐺, 𝐺′ ). □ 3. convex compactness property A locally convex vector space 𝐸 has the convex compactness (cc) property when the closed absolutely convex hull of every compact subset 𝐾, of 𝐸, is compact. This is a well known notion of the theory of locally convex vector spaces which appears in relation to questions dealing with duality and completeness. For example, it is crucial in the characterization of semi-reflexivity for the additive groups of locally convex vector spaces (see [6]) and is equivalent to completeness for metrizable locally convex vector spaces (see [13].) It would be desirable to extend this kind of results to general topological abelian groups but one basic obstruction to accomplish it is the lack of a sufficiently general extension of the Hahn-Banach theorem to this wider context. So it is not possible in general to translate most results that hold for locally convex vector spaces to general topological abelian groups in a simple way as we shall see below. However, we shall show next that with an appropriate modification of several basic tools used in the theory of locally convex vector spaces it is still possible to obtain an extension of that theory to a more general context. ˆ is denoted Here on, given any topological abelian group 𝐺, the duality ⟨𝐺, 𝐺⟩ by p. The group is said locally quasi-convex if it is MAP and admits a base of neighborhood of the identity consisting of 𝑝-convex sets. We say that a locally quasi-convex group 𝐺 has the convex compactness (cc) property when the 𝑝−convex hull of every compact subset of 𝐺 is compact as well (see also [3].) Our first result

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shows that the cc property characterizes completeness for metrizable locally quasiconvex groups what generalizes the analogous result given by Ostling and Wilansky for locally convex vector spaces (cf. [13].) In the sequel we identify 𝕋 to the interval [−1/2, 1/2) in order to use the additive notation. With this notation, given a duality 𝛼 = ⟨𝐺, 𝐺′ ⟩ and any subset 𝐴 in 𝐺, we have that 𝐴𝛼 = {𝑔 ′ ∈ 𝐺′ : ∣⟨𝑔, 𝑔 ′ ⟩∣ ≤ 1/4 for all 𝑔 ∈ 𝐴} Theorem 2. Let 𝐺 be a metrizable locally quasi-convex group. Then 𝐺 is complete if and only if 𝐺 has the cc property. Proof. Necessity is clear. Sufficiency: Let {𝑎𝑛 }𝑛