Group valued null sequences and metrizable non-Mackey groups

Abstract

For a topological abelian group X we topologize the group c 0 .X / of all X-valued null sequences in a way such that when X D R the topology of c 0 .R/ coincides with the usual Banach space topology of the classical Banach space c 0 . If X is a nontrivial compact connected metrizable group, we prove that c 0 .X / is a non-compact Polish locally quasi-convex group with countable dual group c 0 .X /^. Surprisingly, for a compact metrizable X, countability of c 0 .X/^leads to connectedness of X.Our principal application of the above results is to the class of locally quasi-convex Mackey groups (LQC-Mackey groups). A topological group .G; / from a class G of topological abelian groups will be called a Mackey group in G or a G -Mackey group if it has the following property: if is a group topology in G such that .G; / 2 G and .G; / has the same character group as .G; /, then Ä .Based upon the results obtained for c 0 .X/, we provide a large family of metrizable precompact (hence, locally quasi-convex) connected groups which are not LQC-Mackey. Namely, we show that for a connected compact metrizable group X ¤ ¹0º, the group c 0 .X /, endowed with the topology induced from the product topology on X N , is a metrizable precompact connected group which is not a Mackey group in LQC.Since metrizable locally convex spaces always carry the Mackey topology -a wellknown fact from Functional Analysis -, our results prove that a Mackey theory for abelian groups is not a simple traslation of items known to hold for locally convex spaces. This paper is a contribution to the Mackey theory for groups, where properties of a topological nature like compactness or connectedness have an important role.

Locally Quasi-Convex Compatible Topologies on a Topological Group

“…of C (G), are metrizable (so, second countable, since G is separable by (3.4) in [23]). This makes it natural to ask whether all compatible topologies are metrizable.…”

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

“…It is easy to check that the collection {V N : V ∈ N (X)} forms a basis at 0 of a group topology on X N . This topology is called the uniform topology and is denoted by u X [9]. Following [9], denote by c 0 (X) the group of all X-valued null sequences, i.e., Abelian topological groups and continuous homomorphisms by the assignment X → F 0 (X) := c 0 (X), u 0 , F 0 (p) (x n ) n∈N = p(x n ) n∈N .…”

“…The groups of the form F 0 (X) were thoroughly studied in [9,17]. Our interest in the study of the functor F 0 is explained by the following arguments.…”

“…Our interest in the study of the functor F 0 is explained by the following arguments. Firstly, F 0 preserves many important topological properties as (sequential) completeness, metrizability, separability, maximal almost periodicity, local quasi-convexity and connectedness [9]. Secondly, the group F 0 (T) plays a significant role in the theory of characterized subgroups of compact Abelian groups (see [14,16]).…”

“…Fourthly, since the groups (c 0 (X), p) and (X N , p) have the same group of continuous characters for every Abelian topological group X by [2, 3.6.12], Theorem 1 of [13] shows that the groups F 0 (T) and (c 0 (T), p) have the same countable group Z (N) of continuous characters. More generally, for every metrizable and connected compact Abelian group C, the groups F 0 (C) and (c 0 (C), p) have the same countable group of continuous characters [9] (see also Theorem 3.8 below). This means that the connected precompact metrizable Abelian group (c 0 (C), p) is not Mackey (for the definition of Mackey groups and their basic properties see [7]).…”