Pontryagin reflexive groups are not determined by their continuous characters (Q1288314)

Let \(G\) be an abelian topological group. Let \(\tau\) denote the topology of \(G\) and \(\tau_\omega\) the weak topology induced by all continuous characters of \(G\). The group \(G\) endowed with the topology \(\tau_\omega\) is denoted by \(G_\omega\). \textit{I. Glicksberg} [Can. J. Math. 14, 269-276 (1962; Zbl 0109.20001)] proved: (i) \(G\) and \(G_\omega\) have the same compact subsets. (ii) The topology of \(G\) is determined by the set of all its continuous characters. \textit{R. Venkataraman} [Pac. J. Math. 57, 591-595 (1975; Zbl 0308.22009)] asserted that (i) and (ii) hold for reflexive groups. The authors show that (ii) does not hold for reflexive groups. They provide sufficient conditions on reflexive groups for (ii) to hold. That (i) is false for reflexive groups is the content of \textit{D. Remus} and \textit{F. J. Trigos-Arrieta} [Proc. Am. Math. Soc. 117, 1195-1200 (1993; Zbl 0826.22002)].