×

A theorem on cardinal numbers associated with \({\mathcal L}_\infty\) Abelian groups. (English) Zbl 1037.22001

N. Th. Varopoulos [Proc. Camb. Philos. Soc. 60, 465–516 (1964; Zbl 0161.11103)] defined an \((\mathcal L_{\infty})\)-group to be a topological group in which the topology is the intersection of a decreasing sequence of Hausdorff locally compact group topologies.
J. B. Reade [Proc. Camb. Philos. Soc. 61, 69–74 (1965; Zbl 0136.29604)] subsequently proved that given two different \(\mathcal{L}_{\infty}\) topologies \(\mathcal{L}_1\subset \mathcal{L}_2\) on an Abelian group the character groups \((G,\mathcal{ L}_i)^{\wedge}\) satisfy the relation \[ \left| \frac{(G,{\mathcal L}_2)^{\wedge}}{ (G,{\mathcal L}_1)^{\wedge}}\right| \geq 2^{\aleph_1}. \] Since LCA groups were known to satisfy the stronger (in the absence of the continuum hypothesis) relation \[ \left| \frac{(G,{\mathcal L}_2)^{\wedge}}{ (G,{\mathcal L}_1)^{\wedge}}\right| \geq 2^{\mathfrak{c}},\tag{1} \] Reade conjectured that this relation should also hold for \({\mathcal L}_{\infty}\)-groups.
The purpose of the present paper is to prove that conjecture. When \(\mathcal{L}_1\subset \mathcal{L}_2\) are two different \({\mathcal L}_{\infty}\) topologies on an Abelian group such that \((G,{\mathcal L}_1)\) is separable and \((G,{\mathcal L}_2)^{\wedge}\) is metrizable (in the compact-open topology) the author manages to find a subset \(K\) of \((G,{\mathcal L}_2)^{\wedge}\) which is compact as a subset of \(G_d^\wedge\) (the compact group of all characters of \(G\) with pointwise convergence) yet has a subset \(B\) whose closure in \(G_d^\wedge\) has cardinality \(2^{\mathfrak{c}}\). \(B\) is actually an \(I_0\)-set, a set of interpolation for the almost periodic functions, in the group \((G,{\mathcal L}_1)^{\wedge}\). The proof of this fact is based on Rosenthal’s \(\ell^1\)-theorem and proves inequality (1) for the above groups (\(| (G,{\mathcal L}_1)^\wedge| \leq \mathfrak{c}\), since \((G,{\mathcal L}_1)\) is separable). The general case follows after appropriately combining the structure theory of \({\mathcal L}_{\infty}\) groups developed in N. Th. Varopoulos [loc. cit], L. J. Sulley [J. Lond. Math. Soc., II. Ser. 5, 629–637 (1972; Zbl 0243.22002)] and R. Venkataraman [Monatsh. Math. 100, 47–66 (1985; Zbl 0563.43005)], see also the author and the reviewer [Fund. Math. 159, 195–218 (1999; Zbl 0934.22008)].

MSC:

22A05 Structure of general topological groups
PDFBibTeX XMLCite
Full Text: DOI