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The concept of boundedness and the Bohr compactification of a MAP abelian group. (English) Zbl 0934.22008

Let $G$ denote an Abelian topological group with sufficiently many continuous characters, ${G}^{\wedge }$ the character group of $G$. The Bohr topology on $G$ is the smallest topology that makes all characters of ${G}^{\wedge }$ continuous. By a standard notation ${G}^{+}$ denotes $G$ equipped with the Bohr topology and $bG$ the Bohr compactification (i.e. the closure of ${G}^{+}$ canonically embedded in the product of circles ${T}^{{G}^{\wedge }}\right)$. The notion of “boundedness” for topological Abelian groups was introduced by Vilenkin. Hejcman fixed what could be a canonical boundedness for a topological Abelian group. The authors of the paper under review have introduced the concepts of bounded covering number of a subset $A\subset G$ and of separated boundedness which turn out very important to obtain the subsequent results.

They first provide examples of groups whose canonical boundedness is separated, like ${ℒ}_{\infty }$-groups, DF-spaces (in particular, normed spaces) and the products $E×{G}_{0}$, where $E$ is a DF-space and ${G}_{0}$ a group that contains an open subgroup which is the strict inductive limit of a sequence of compact groups. For further reference let us call $ℋ$ the class formed by the latter. The authors succeed to extend to $ℋ$ a result obtained by van Douwen for discrete groups which, roughly speaking, consists in finding big discrete subsets of ${G}^{+}$ which are $C$-embedded in ${G}^{+}$ and ${C}^{*}$-embedded in $bG$. Namely, they prove that for every subset $A\subset G$ with $G\in ℋ$ there exists a subset $B\subset A$ with cardinality the bounded covering number of $A$, such that $B$ is relatively discrete in ${G}^{+}$, $C$-embedded in ${G}^{+}$ and ${C}^{*}$-embedded in $bG$.

In the second part of the paper, using the deep results already obtained in the first part, the authors generalize the following theorem of Glicksberg: “For a locally compact Abelian group (LCA) $G$, the compact subsets of $G$ and of ${G}^{+}$ coincide.” Previous generalizations of this beautiful result were done by D. Remus and F. J. Trigos-Arrieta who named this property “to respect compactness” and proved [Proc. Am. Math. Soc. 117, 1195-1200 (1993; Zbl 0826.22002)] that Montel spaces are exactly those locally convex reflexive real spaces which respect compactness. Also W. Banaszczyk and E. Martín-Peinador [Annals of the New York Academy of Sciences, Vol. 788, 34-39 (1996)] proved that all nuclear groups respect compactness. The class of nuclear groups, defined by W. Banaszczyk [Additive subgroups of topological vector spaces. Lect. Notes Math. 1466 (Berlin etc. 1991; Zbl 0743.46002)], contains the locally compact Abelian groups and the nuclear locally convex spaces. Finally, W. W. Comfort, F. J. Trigos-Arrieta and T. S. Wu [Fund. Math. 143, 119-136 (1993; Zbl 0812.22001)] proved that LCA groups verify a property stronger than respecting compactness, and left as an open question which other classes of groups would also “strongly respect compactness” (src).

In the paper under review it is proved that Montel spaces src, and can be characterized by this property in the realm of locally convex reflexive spaces; that ${ℒ}_{\infty }$-groups src, and also preservation of compactness for free abelian groups is considered. The paper is very well written and contains 32 bibliographic items.

Note: We know that these results have been carried on. The first author has proved that nuclear groups have separated boundedness, and that complete nuclear groups src (to appear in Houston J. Math.). The second author has proved that, for metrizable groups, the properties of respecting compactness, src and to satisfy the van Douwen theorem are all equivalent.

##### MSC:
 22A05 Structure of general topological groups 54C45 $C$- and ${C}^{*}$-embedding 46A99 Topological linear spaces 54H11 Topological groups (topological aspects) 46A04 Locally convex Fréchet spaces, etc.