Free Abelian topological groups and the Pontryagin-Van Kampen duality

Abstract

We study the class of Tychonoff topological spaces such that the free Abelian topological group A(X) is reflexive (satisfies the Pontryagin-van Kampen duality). Every such X must be totally path-disconnected and (if it is pseudocompact) must have a trivial first cohomotopy group π1(X). If X is a strongly zero-dimensional space which is either metrisable or compact, then A(X) is reflexive.

Pontryagin duality for metrizable groups

“…We prove that completeness is a necessary but not sufficient condition for the Pontryagin reflexivity of a metrizable group. With this result we answer the question posed recently by Pestov (see [11]), about the Pontryagin reflexivity of complete, C Ï ech complete topological groups. We also conclude from Theorem 1 that a separable completely metrizable topological group is Pontryagin reflexive if and only if the canonical mapping a G is an algebraic isomorphism.…”

“…l p Y 0`p`1), there are complete, C Ï ech complete topological groups which are not Pontryagin reflexive. This answers negatively the question posed by Pestov in [11].…”

Locally minimal topological groups 2

“…2 (B) For a compact Hausdorff space K , the free abelian topological group A(K ) is a hemicompact k-space, and a cobasis for the compact sets is given by the family of sets {η(K )+ n · · · +η(K ): n ∈ N} [24]. (C) For a compact Hausdorff space K , the dual group of A(K ) is topologically isomorphic to C(K , T); more precisely, [29]. In particular A(K ) ∧ is a UFSS group (according to Example 6.9).…”

Characterizing group C*-algebras through their unitary groups: the Abelian case

“…When D is a totally disconnected compact group, the only continuous characters of C(D, T ) are linear combinations with coefficients in Z of evaluations of elements of D, i.e., the group C(D, T ) is isomorphic to the free Abelian group A(D) on D [13] (see [9] for more on the duality between C(X , T ) and A(X ) based on the exact sequence in Lemma 4.1).…”