Locally quasi-convex convergence groups (Q2216665)

This paper provides a definition of local quasi-convexity for abelian convergence groups, that generalizes the corresponding notion for abelian topological groups. The very simple notion of quasi-convex subset of an abelian topological group, introduced by Vilenkin in 1951, was a seminal tool to develop Pontryagin duality beyond the locally compact abelian groups. Further, from the consideration of locally quasi-convex groups as a class containing the locally convex spaces, many well-known theorems of Functional Analysis that hold for locally convex spaces, have been generalized to the wider class of locally quasi-convex groups, obtaining at least partial versions of them for abelian groups. In the paper under review some consequences of the given definition are obtained, which point out to a deeper study of the so called \(c\)-reflexivity. The \(c\)-reflexivity leans on the continuous convergence structure, which roughly speaking, plays the role of the compact open topology in Pontryagin duality. For locally compact abelian topological groups these two notions are equivalent. However, the continuous convergence structure is valid also for the wider class of convergence abelian groups. The tool of local quasi-convexity introduced by Sharma might be very useful to study \(c\)-reflexive convergence groups. The first part of the paper deals with the homeomorphism group of the rationals, \(H(\mathbb{Q})\) endowed with a convergence structure, which turns out to be the coarsest admissible convergence structure that makes \(H(\mathbb{Q})\) a convergence group. The result obtained is a particular case of Theorem 5 in [\textit{W. R. Park}, Math. Ann. 199, 45--54 (1972; Zbl 0231.54006)], where it is done for the group of homeomorphisms \(H(X)\), with \(X\) a convergence group.