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

N. Th. Varopoulos [Proc. Camb. Philos. Soc. 60, 465–516 (1964; Zbl 0161.11103)] defined an $\left({ℒ}_{\infty }\right)$-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 ${ℒ}_{\infty }$ topologies ${ℒ}_{1}\subset {ℒ}_{2}$ on an Abelian group the character groups ${\left(G,{ℒ}_{i}\right)}^{\wedge }$ satisfy the relation

$\left|\frac{{\left(G,{ℒ}_{2}\right)}^{\wedge }}{{\left(G,{ℒ}_{1}\right)}^{\wedge }}\right|\ge {2}^{{\aleph }_{1}}·$

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

$\left|\frac{{\left(G,{ℒ}_{2}\right)}^{\wedge }}{{\left(G,{ℒ}_{1}\right)}^{\wedge }}\right|\ge {2}^{𝔠},\phantom{\rule{2.em}{0ex}}\left(1\right)$

Reade conjectured that this relation should also hold for ${ℒ}_{\infty }$-groups.

The purpose of the present paper is to prove that conjecture. When ${ℒ}_{1}\subset {ℒ}_{2}$ are two different ${ℒ}_{\infty }$ topologies on an Abelian group such that $\left(G,{ℒ}_{1}\right)$ is separable and ${\left(G,{ℒ}_{2}\right)}^{\wedge }$ is metrizable (in the compact-open topology) the author manages to find a subset $K$ of ${\left(G,{ℒ}_{2}\right)}^{\wedge }$ 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 ${2}^{𝔠}$. $B$ is actually an ${I}_{0}$-set, a set of interpolation for the almost periodic functions, in the group ${\left(G,{ℒ}_{1}\right)}^{\wedge }$. The proof of this fact is based on Rosenthal’s ${\ell }^{1}$-theorem and proves inequality (1) for the above groups ($|{\left(G,{ℒ}_{1}\right)}^{\wedge }|\le 𝔠$, since $\left(G,{ℒ}_{1}\right)$ is separable). The general case follows after appropriately combining the structure theory of ${ℒ}_{\infty }$ 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