Eberlein–šmulyan Theorem for Abelian Topological Groups

Bruguera, Montserrat; Martín-Peinador, Elena; Tarieladze, V.

doi:10.1112/s0024610704005629

Cited by 8 publications

(6 citation statements)

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“…(v)⇒(vi) is clear (see also [11]). (vi)⇒(i), (vii)⇒(i), (viii)⇒(i), (ix)⇒(i), (x)⇒(i) Let X ∼ = R n × X 0 , where n ≥ 0 and X 0 has an open compact subgroup K [48, 24.30].…”

Section: Proof Of Theorem 110mentioning

confidence: 77%

“…Applying [6, 2.4] once again we obtain that F 0 (L) is a real locally convex space. Proposition 3.3, Fact 1.6(i) and [11,Corollary 16] immediately imply:…”

Section: Example 218mentioning

confidence: 88%

“…The Schur property in Abelian topological groups was intensively studied by Martín-Peinador and Tarieladze in [51] (see also [36,46]). Many other results concerning preservation and respectness of topological properties under the Bohr functor see [11]. A complete characterization of Banach spaces which have the Schur property is given in [46].…”

Section: Proof Of Theorem 110mentioning

confidence: 99%

“…Clearly, compact and sequentially compact subsets are countably compact. On the other hand, there are sequentially compact subsets of a completely regular Hausdorff space which are nonclosed, and their closure are not countably compact (see Examples 28 and 29 in[11]). So topological spaces, in which any two of these in general different topological properties coincide, are of independent interest.…”

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confidence: 99%

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Topological properties of the group of the null sequences valued in an Abelian topological group

Gabriyelyan

2016

Topology and its Applications

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Following [23], denote by F0 the functor on the category TAG of all Hausdorff Abelian topological groups and continuous homomorphisms which passes each X ∈ TAG to the group of all X-valued null sequences endowed with the uniform topology. We prove that if X ∈ TAG is an (E)-space (respectively, a strictly angelic space or aŠ-space), then F0(X) is an (E)-space (respectively, a strictly angelic space or aŠspace). We study respected properties for topological groups in particular from categorical point of view. Using this investigation we show that for a locally compact Abelian (LCA) group X the following are equivalent: 1) X is totally disconnected, 2) F0(X) is a Schwartz group, 3) F0(X) respects compactness, 4) F0(X) has the Schur property. So, if a LCA group X has non-zero connected component, the group F0(X) is a reflexive non-Schwartz group which does not have the Schur property. We prove also that for every compact connected metrizable Abelian group X the group F0(X) is monothetic that generalizes a result by Rolewicz for X = T.2000 Mathematics Subject Classification. Primary 22A10, 22A35, 43A05; Secondary 43A40, 54H11. Key words and phrases. group of null sequences, locally compact group, monothetic group, (E)-space, strictly angelic space, S-space, respect compactness, the Schur property. 1 2 S.S. GABRIYELYAN X + := (X, τ + ). Note that B is finitely multiplicative. It is well-known that the groups X and X + have the same set of continuous characters, and B(X) = X if and only if X is precompact (see [1,16]).It is known that, on the class LCA the Bohr functor B preserves connectedness [67], covering dimension [44,67] and realcompactness [14]. On the other hand, X + is normal if and only if X is σ-compact [68]. Note also that, if X is discrete and infinite, then X + is sequentially complete [28] and non-pseudocompact [18]. The same holds for the general case: if X is a non-compact LCA group, then X + is sequentially complete and non-pseudocompact [22].(C) Respected properties. Using the Bohr functor we can define respected properties in subcategories of MAPA. Following [57], if P denotes a topological property, then we say that a M AP A group X respects P if X and X + have the same subspaces with P. Let A A A be a subcategory of MAPA.We say that P is a respected property in A A A if every X ∈ A A A respects P.Problem 1.4. Which topological properties are respected for a given subcategory A A A of MAPA?Recall that X ∈ MAPA is said to have: 1) the Glicksberg property or X respects compactness if the compact subsets of X and X + coincide, and 2) the Schur property or X respects sequentiality if X and X + have the same set of convergent sequences. Clearly, if X respects compactness, then it has the Schur property.Let X ∈ LCA and let P be a topological property. The question of whether X respects P has been intensively studied for many natural properties P. The famous Glicksberg theorem [40] states that every LCA group X respects compactness, and hence X has the Schur property. So the Glicksberg and the Schur prope...

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Section: Proof Of Theorem 110mentioning

confidence: 77%

“…Applying [6, 2.4] once again we obtain that F 0 (L) is a real locally convex space. Proposition 3.3, Fact 1.6(i) and [11,Corollary 16] immediately imply:…”

Section: Example 218mentioning

confidence: 88%

Section: Proof Of Theorem 110mentioning

confidence: 99%

mentioning

confidence: 99%

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Topological properties of the group of the null sequences valued in an Abelian topological group

Gabriyelyan

2016

Topology and its Applications

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“…The proof follows from joint consideration of Fact 1 and Case 1. r Theorem 2 easily applies to deduce that a locally compact topological group X is metrizable if C p ðX Þ is separable. [6], where it was shown (under continuum hypothesis (CH)) that a compact group is metrizable if it is sequentially compact. Later, in [22] it was noted that the metrizability of a compact angelic group can be also proved without (CH).…”

Section: Related Results and Proofsmentioning

confidence: 99%

Lindelöf spaces C(X) over topological groups

Ka̧kol

Pellicer²,

Martín-Peinador

et al. 2008

Forum Mathematicum

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Abstract. Theorem 1 proves (among the others) that for a locally compact topological group X the following assertions are equivalent: (i) X is metrizable and s-compact.Þ is a separable metrizable and complete locally convex space. (vi) C c ðX Þ is compactly dominated by irrationals. This result supplements earlier results of Corson, Christensen and Calbrix and provides several applications, for example, it easily applies to show that: (1) For a compact topological group X the Eberlein, Talagrand, Gul'ko and Corson compactness are equivalent and any compact group of this type is metrizable. (2) For a locally compact topological group X the space C p ðX Þ is Lindelö f i¤ C c ðX Þ is weakly Lindelö f. The proofs heavily depend on the following result of independent interest: A locally compact topological group X is metrizable i¤ every compact subgroup of X has countable tightness (Theorem 2). More applications of Theorem 1 and Theorem 2 are provided.

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Maximally Almost Periodic Groups and Respecting Properties

Gabriyelyan

2019

Descriptive Topology and Functional Analysis II

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For a Tychonoff space X, denote by P the family of topological properties P of being a convergent sequence or being a compact, sequentially compact, countably compact, pseudocompact and functionally bounded subset of X, respectively. A maximally almost periodic (M AP ) group G respects P if P(G) = P(G + ), where G + is the group G endowed with the Bohr topology. We study relations between different respecting properties from P and show that the respecting convergent sequences (=the Schur property) is the weakest one among the properties of P. We characterize respecting properties from P in wide classes of M AP topological groups including the class of metrizable M AP abelian groups. Every real locally convex space (lcs) is a quotient space of an lcs with the Schur property, and every locally quasi-convex (lqc) abelian group is a quotient group of an lqc abelian group with the Schur property. It is shown that a reflexive group G has the Schur property or respects compactness iff its dual group G ∧ is c0-barrelled or g-barrelled, respectively. We prove that an lqc abelian kω-group respects all properties P ∈ P. As an application of the obtained results we show that (1) the space C k (X) is a reflexive group for every separable metrizable space X, and (2) a reflexive abelian group of finite exponent is a Mackey group. P 0 := {S, C, SC, CC, PC} and P := P 0 ∪ {FB}.Let G be a maximally almost periodic (M AP ) topological group G (for all relevant definitions, see Section 2). We denote by G + the group G endowed with the Bohr topology. Following [57], a M AP group G respects a topological property P if P(G) = P(G + ).The famous Glicksberg theorem [36] states that every locally compact abelian (LCA) group respects compactness. If a M AP group G respects compactness we shall say also that G has the Glicksberg property. Trigos-Arrieta [60,61] proved that countable compactness, pseudocompactness and functional boundedness are respected by LCA groups. Banaszczyk and Martín-Peinador [9] generalized these results to all nuclear groups. Nuclear groups were introduced and thoroughly studied by Banaszczyk in [8]. The concept of Schwartz topological abelian groups is appeared in [4]. This notion generalizes the well-known notion of a Schwartz locally convex space. All nuclear groups are Schwartz groups [4]. Außenhofer [2] proved that every locally quasi-convex Schwartz group respects compactness. For a general and simple approach to the theory of properties respected by M AP topological groups see [24].Let (E, τ ) be a locally convex space (lcs for short), E ′ the dual space of E and let τ w = σ(E, E ′ ) be the weak topology on E. Set E w := (E, τ w ). An lcs E is said to have the Schur property if E and E w have the same convergent sequences, i.e., S(E) = S(E w ). Considering E as an additive 2000 Mathematics Subject Classification. Primary 22A05, 46A03; Secondary 54H11.

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Eberlein–šmulyan Theorem for Abelian Topological Groups

Cited by 8 publications

References 20 publications

Topological properties of the group of the null sequences valued in an Abelian topological group

Topological properties of the group of the null sequences valued in an Abelian topological group

Lindelöf spaces C(X) over topological groups

Maximally Almost Periodic Groups and Respecting Properties

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