Some aspects of nuclear vector groups (Q2717711)

Nuclear groups were defined by \textit{W. Banaszczyk} [Lect. Notes Math. 1466 (Berlin etc. 1991; Zbl 0743.46002)] as a Hausdorff variety of Abelian topological groups containing locally compact groups and additive groups of nuclear vector spaces. The natural question as to whether this is the variety generated by these groups is addressed in this paper. NEWLINENEWLINENEWLINEThe author considers three varieties of topological groups related to nuclear groups: the variety \(\mathcal{V}_0\) generated by nuclear vector spaces, the variety \(\mathcal{V}_1\) generated by nuclear vector spaces and discrete Abelian groups and the variety \(\mathcal{V}_2\) consisting of all nuclear groups. All three varieties are proved to be different. NEWLINENEWLINENEWLINEThe variety \(\mathcal{V}_0\) is shown to be properly contained in \(\mathcal{V}_1\) by proving that the cardinality of discrete groups in \(\mathcal{V}_1\) is at most that of the real numbers.NEWLINENEWLINENEWLINEThe key tool in proving that \(\mathcal{V}_1\) is properly contained in \(\mathcal{V}_2\) is a group of sequences named \(\Sigma_0\) containing the usual space of rapidly decreasing sequences as its connected component. It is a well-known theorem of \textit{T. Komura} and \textit{Y. Komura} [Math. Ann. 162, 284-288 (1966; Zbl 0156.13402)] that the latter space is a universal generator for the class of locally convex nuclear spaces. The author mimics the proof of this fact in showing that the group \(\Sigma_0\) is a universal generator for the class of nuclear groups. The paper ends by proving that this group is not in the variety \(\mathcal{V}_1\), a fact which reveals in quite a precise manner the disposition of the variety \(\mathcal{V}_1\) within \(\mathcal{V}_2\).