Group valued null sequences and metrizable non-Mackey groups

Abstract

Abstract. For a topological abelian group X we topologize the group c 0 ( X ) $c_0(X)$ of all X-valued null sequences in a way such that when X = ℝ ${X={\mathbb {R}}}$ the topology of c 0 ( ℝ ) $c_0({\mathbb {R}})$ coincides with the usual Banach space topology of the classical Banach space c0 . If X is a non-trivial compact connected metrizable group, we prove that c 0 ( X ) $c_0(X)$ is a non-compact Polish locally quasi-convex group with countable dual group c 0 ( X ) ∧ $c_0(X)^{\wedge }$ . Surprisingly, for a compact metrizable X, countability of c 0 ( X ) ∧ $c_0(X)^{\wedge }$ leads to connectedness of X. Our principal application of the above results is to the class of locally quasi-convex Mackey groups ( LQC $\rm {LQC}$ -Mackey groups). A topological group ( G , μ ) $(G,\mu )$ from a class 𝒢 $\mathcal {G}$ of topological abelian groups will be called a Mackey group in 𝒢 $\mathcal {G}$ or a 𝒢 $\mathcal {G}$ -Mackey group if it has the following property: if ν is a group topology in G such that ( G , ν ) ∈ 𝒢 ${(G,\nu )\in \mathcal {G}}$ and ( G , ν ) $(G,\nu )$ has the same character group as ( G , μ ) $(G,\mu )$ , then ν ≤ μ ${\nu \le \mu }$ . Based upon the results obtained for c 0 ( X ) $c_0(X)$ , we provide a large family of metrizable precompact (hence, locally quasi-convex) connected groups which are not LQC $\rm {LQC}$ -Mackey. Namely, we show that for a connected compact metrizable group X ≠ { 0 } ${X\ne \lbrace 0\rbrace }$ , the group c 0 ( X ) $c_0(X)$ , endowed with the topology induced from the product topology on X ℕ $X^{{\mathbb {N}}}$ , 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 well-known 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.