Pseudocompactness on groups.

*(English)*Zbl 0777.22003
General topology and applications, Proc. 5th Northeast Conf., New York/NY (USA) 1989, Lect. Notes Pure Appl. Math. 134, 369-378 (1991).

[For the entire collection see Zbl 0744.00027.]

Let \(G\) be a locally compact Abelian group with character group \(G^ \wedge\). The author denotes by \(G^ +\) the underlying group \(G\) endowed with the weakest topology making every character from \(G^ \wedge\) continuous. It is a theorem of I. Glicksberg [Can. J. Math. 14, 269-276 (1962; Zbl 0109.020)] that \(G\) and \(G^ +\) have the same compact subsets. In this paper, a new proof of Glicksberg’s theorem is given without using tools from functional analysis. The case if \(G\) is discrete was treated by W. W. Comfort and the author [Lect. Notes Pure Appl. Math. 134, 25-33 (1991; see the preceding review Zbl 0777.22002)]. It is a key result for the proof of the general theorem given in this paper. Finally the author shows that \(G\) and \(G^ +\) have the same pseudocompact subsets.

{Reviewer’s remark: Further results concerning relationships between \(G\) and \(G^ +\) can be found in the author’s paper [J. Pure Appl. Algebra 70, No. 1/2, 199-210 (1991; Zbl 0724.22003)]}.

Let \(G\) be a locally compact Abelian group with character group \(G^ \wedge\). The author denotes by \(G^ +\) the underlying group \(G\) endowed with the weakest topology making every character from \(G^ \wedge\) continuous. It is a theorem of I. Glicksberg [Can. J. Math. 14, 269-276 (1962; Zbl 0109.020)] that \(G\) and \(G^ +\) have the same compact subsets. In this paper, a new proof of Glicksberg’s theorem is given without using tools from functional analysis. The case if \(G\) is discrete was treated by W. W. Comfort and the author [Lect. Notes Pure Appl. Math. 134, 25-33 (1991; see the preceding review Zbl 0777.22002)]. It is a key result for the proof of the general theorem given in this paper. Finally the author shows that \(G\) and \(G^ +\) have the same pseudocompact subsets.

{Reviewer’s remark: Further results concerning relationships between \(G\) and \(G^ +\) can be found in the author’s paper [J. Pure Appl. Algebra 70, No. 1/2, 199-210 (1991; Zbl 0724.22003)]}.

Reviewer: D.Remus (Hagen)