Characterizable groups: Some results and open questions (Q429329)

Let \(G\) be a topological group and let \( \mathbf{u} =(u_n)_{n\in\mathbb N}\) be a sequence in its dual group \(G^\wedge\). The subgroup \(s_{\mathbf{u}}(X):=\{x\in G:\;u_n(x)\to 0\}\) is said to be the characterized subgroup of \(X\) by the sequence \({\mathbf{u}} \). An abelian Polish group \(G\) is called characterizable if there is a continuous monomorphism \(p:G\to X\) with dense image where \(X\) is a compact metrizable group such that \(p(G)=s_{\mathbf{u}}(X)\) for a suitable sequence \(\mathbf{u}\) in \(X^\wedge\).NEWLINENEWLINEThe main result of this article states that every second countable locally compact abelian group \(G\) is characterizable. As a consequence, the author obtains that every second countable locally compact abelian group is algebraically the character group of a complete countable MAP group. Further, a characterizable group of finite exponent is necessarily locally compact and therefore Pontryagin-reflexive.NEWLINENEWLINEBesides these results, a lot of interesting questions are formulated, e.g. to determine all characterizable groups. The question whether every characterizable group is Pontryagin-reflexive is posed as well as the question, which permanence properties hold for the class of characterizable groups.