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

Abstract

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...

Locally convex spaces and Schur type properties

“…Proof. (i) If A is not precompact, Theorem 5 of [3] implies that A has an infinite uniformly discrete subset C, i.e., there is a neighborhood U of zero in E such that c−c ′ ∈ U for every distinct c, c ′ ∈ C. Thus, by Lemma 2.1 of [19], C and hence also A are not functionally bounded, a contradiction.…”

Maximally Almost Periodic Groups and Respecting Properties

“…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 ).…”

On reflexivity of the group of the null sequences valued in an Abelian topological group

“…Thirdly, Rolewicz [26] observed that the complete metrizable group F 0 (T) is monothetic (see a proof of this fact in [11, pp. 20-21] or in [18]). So monothetic Polish groups need not be compact nor discrete.…”

“…In particular, the last two arguments show that the groups of the form F 0 (X) represent a nice source of (counter)examples in different areas of topological algebra. For other important properties of the functor F 0 see [18].…”

“…Indeed, if B is a Banach space, then F 0 (B) is also a Banach space by [18]. Hence F 0 (B) considered as an Abelian topological group is reflexive [27].…”