*(English)*Zbl 1037.22001

*N. Th. Varopoulos* [Proc. Camb. Philos. Soc. 60, 465–516 (1964; Zbl 0161.11103)] defined an $\left({\mathcal{L}}_{\infty}\right)$-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

Since LCA groups were known to satisfy the stronger (in the absence of the continuum hypothesis) relation

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 ${2}^{\U0001d520}$. $B$ 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 ($|{(G,{\mathcal{L}}_{1})}^{\wedge}|\le \U0001d520$, 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 |