Mackey groups and Mackey topologies
Volume 567 / 2021
Abstract
Inspired by the well-known Mackey–Arens Theorem from functional analysis, Chasco, Martín Peinador and Tarieladze (1999) introduced the notions of Mackey groups and Mackey topologies in the realm of locally quasi-convex topological abelian groups. In analogy with the case of locally convex spaces, the problem is to decide whether the poset $\mathcal{C}(G)$ of all compatible topologies for a locally quasi-convex $G$ has a top element, namely, the Mackey topology ($G$ is a Mackey group, if this is the original topology of $G$). The problem turned out to be unusually hard, only recently it was shown that unlike the case of locally convex spaces, treated by the Mackey–Arens Theorem, some locally quasi-convex groups need not have a Mackey topology.
This survey provides a self-contained exposition of the historical background of the problem and its recent solution; it also provides a large variety of locally quasi-convex groups for which the Mackey topology exists. The major source of such groups are the $g$-barreled groups, introduced by Chasco et al. (1999). We gather a large collection of sufficient conditions which ensure $g$-barreledness. Among them the most prominent ones seem to be various levels of compactness/completeness-like properties, as local compactness, pseudocompactness, Čech-completeness, Namioka property and Baire property (although, this property alone is not sufficient to guarantee $g$-barreledness).
In the case of locally convex vector spaces the Mackey topology can be described explicitly as a uniform convergence topology on a family of subsets of the dual space. Following this idea, the Arens topology $\boldsymbol{\tau}_{\rm qc}(G)$ of a locally quasi-convex group $G$ was defined similarly by Chasco et al. (1999); the groups $G$ having compatible Arens topology are called Arens groups. The first example of a non-Arens group was given by Bonales, Trigos-Arrieta and Vera Mendoza (2003). We provide a large class of non-Arens groups; many of those are Mackey groups. This also motivates the study of the class of those groups that are simultaneously Mackey and Arens, called Mackey–Arens groups here. A prominent example of Mackey–Arens groups are the $g$-barreled groups.
Following the ideas of Barr and Kleisli (2001) we also provide a categorical approach to the Mackey topology problem, which allows us to obtain, in particular, a complete solution in the class of linearly topologized groups, nuclear groups, etc. In the case of the category of linearly topologized groups we also provide an explicit description of the Mackey topology. This has various applications in the case of bounded locally quasi-convex groups (e.g., abelian precompact bounded Baire groups are Mackey, etc.).
Finally, we dedicate attention also to the size and “shape” of the poset $\mathcal{C}(G)$ of all compatible topologies, with particular emphasis on the LCA case: while $|\mathcal{C}(G)|=1$ for a compact group $G$, for a non-compact LCA groups $G$ the complete lattice $\mathcal{C}(G)$ has cardinality $\ge \mathfrak{c}$ and in case $G$ is not $\sigma$-compact it is as large as possible and can be described in a satisfactory way.
