The Mackey-Arens theorem states that for a locally convex vector space \((E,\tau)\), there exists a finest locally convex vector space topology \(\mu\) on \(E\) such that the dual spaces \((E,\tau)'\) and \((E,\mu)'\) are equal. In this situation, \((E,\mu)\) is called a Mackey space and \(\mu\) is called Mackey topology on \(E\).
The class of locally quasi-convex (lqc) groups is an analogue for abelian Hausdorff groups of locally convex spaces. Two lqc group topologies \(\tau_1,\tau_2\) on an abelian group \(G\) are called compatible if they induce the same set of continuous characters, i.e., the character groups \((G,\tau_1)^\wedge\) and \((G,\tau_2)^\wedge\) are algebraically isomorphic.
For a lqc group topology \(\tau\) on an abelian group \(G\), let \(\mathcal{C}(G,\tau)\) denote the set of all group topologies compatible with \(\tau\). Unlike in the realm of locally convex spaces, this set has in general no top element. In the case that such a top element exists, it is called Mackey group topology and the group is called a Mackey group. The study of Mackey group topologies was initiated by M. J. Chasco et al. [Stud. Math. 132, No. 3, 257–284 (1999; Zbl 0930.46006)]. In that article it was already observed that the (necessarily unique) Mackey topology exists if and only if the supremum of two compatible group topologies is again compatible; this was described by means of \( \mathfrak{S}\)-topologies.
In the first part of the present paper, the author introduces a class \(\mathrm{Mac}\) of \(\sigma(G^\wedge,G)\)-compact subsets of \((G,\tau)^\wedge\) which have an additional property described by means of quasi-convex sets. Equivalently, a \(\sigma(G^\wedge,G)\)-compact set \(K\) belongs to \(\mathrm{Mac}\) if and only if the supremum of the weak topology \(\sigma(G,G^\wedge)\) and the topology of uniform convergence \(\tau_K\) on \(K\) is compatible with \(\tau\). The author starts a very careful study of the class \(\mathrm{Mac}\) and gives sufficiency conditions when for example the product of two Mackey groups is again a Mackey group by means of \(\mathrm{Mac}\).
In the second part of the paper, the author studies locally convex spaces. As he had shown before, a Mackey space topology need not be a Mackey group topology. However, in a diagram, the relation between Mackey spaces and Mackey groups as well as sufficiency conditions are clearly presented. The author gives a characterization, using the class \(\mathrm{Mac}\) introduced above, to state when a locally convex Mackey space is a Mackey group. The paper closes with the question whether certain classes of Mackey spaces (e.g., quasi-barrelled spaces) have a Mackey group topology.