Group valued null sequences and metrizable non-Mackey groups

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 = 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 non-trivial 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)(boolean AND). Surprisingly, for a compact metrizable X, countability of c(0)(X)(boolean AND) 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, mu) 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 nu is a group topology in G such that (G, nu) is an element of G and (G, nu) has the same character group as (G, mu), then nu <= mu. 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 not equal {0}, 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 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.