Group valued null sequences and metrizable non-Mackey groups

Abstract

For a topological abelian group X we topologize the group c0(X) of all X-valued null sequences in a way such that when X= the topology of c0() 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 c0(X) is a non-compact Polish locally quasi-convex group with countable dual group c0(X). Surprisingly, for a compact metrizable X, countability of c0(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 of topological abelian groups will be called a Mackey group in or a -Mackey group if it has the following property: if is a group topology in G such that (G,) and (G,) has the same character group as (G,), then . Based upon the results obtained for c0(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 X0, the group c0(X), endowed with the topology induced from the product topology on X, 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.