Dieter Remus

Abelian groups which satisfy Pontryagin duality need not respect compactness

Let G be a topological Abelian group with character group GA. We will say that G respects compactness if its original topology and the weakest topology that makes each element of GA continuous produce the same compact subspaces. We show the existence of groups which satisfy Pontryagin duality and do not respect compactness, thus furnishing counterexamples to a result published by Venkataraman in 1975. Our counterexamples will be the additive groups of all reflexive infinite-dimensional real Banach spaces. In order to do so, we first characterize those locally convex reflexive real spaces whose additive groups respect compactness. They are exactly the Montel spaces. Finally, we study the class of those groups that satisfy Pontryagin duality and respect compactness. 0. INTRODUCTION AND NOTATION Let (G, z) be a topological Abelian group with underlying group G and topology T. A character of G is a homomorphism from G into the circle group T. Denote by (G, r)A the group of continuous characters of (G, T) with multiplication defined pointwise, equipped with the compact-open topology. We will say that (G, T) is a maximally almost periodic group if, for every a e G different from the identity, there exists X E (G, T)A such that X(a)

Complete minimal and totally minimal groups

1. If (G, T) is indeed maximally almost periodic, we will say that (G, T) satisfies Pontryagin duality if the function t: (G, T) -* (G, T)AA, defined by t(x)(x) = x(x) for all x E G and X E (G, T)A, is a topological isomorphism. Let LCA, g, and MAP denote the classes of those groups that are locally compact, satisfy Pontryagin duality, and are maximally almost periodic, respectively. The classical theorem on duality due to Pontryagin and van Kampen states that LCA C p. The class p, however, is strictly wider than LCA: for example, 0 is closed under arbitrary products [6], a property that LCA does not hold. Let (G, T) e MAP. Denote by Tw the weakest topology on G that makes every element in (G, T)A continuous. It follows that (G, T,) is a totally bounded Received by the editors March 18, 1991 and, in revised form, July 20, 1991. 1991 Mathematics Subject Classification. Primary 22A05, 22D35, 46Al 1, 54A10; Secondary 46A20, 46A50. ? 1993 American Mathematical Society 0002-9939/93

Long chains of Hausdorff topological group topologies

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The Bohr topology of Moore groups

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Minimal and precompact group topologies on free groups

Abstract First we construct complete totally minimal topological groups which are not locally compact. Extending and generalizing a construction of S. Dierolf and U. Schwanengel (using semi-direct products) we find then a class A of topological groups such that all products of elements of A are minimal topological groups. This gives further examples of complete minimal topological groups being not locally compact and partial answers to a question posed by A.V. Arhangelskiǐ.

Long chains of topological group topologies—A continuation

Abstract Comfort, W.W. and D. Remus, Long chains of Hausdorff topological group topologies, Journal of Pure and Applied Algebra 70 (1991) 53-72. For certain classes 4 of topological group topologies on a group G, we determine those cardinals B which arise as the length of a chain in 4. We show, for example, that if G is Abelian with /Cl =IY?W and 4 is either the class B of totally bounded (=pre-compact and Hausdorff) topological group topologies for G or the class &’ of non-totally bounded topological group topologies for G, then there is a chain 0 in 4 such that lP/ =2(a’). The same conclusions hold when G is a free group of cardinality c( and 4 = B. Introduction The symbols a, p, y, K and A denote infinite cardinal numbers, and o is the least infinite cardinal number. The symbols c and q denote ordinal numbers. The sym- bols m and QJ denote respectively the class of all infinite cardinal numbers and the class of all ordinal numbers. As usual for . The least cardinal greater than K is written K’. The continuum hypothesis [CH] is the statement W’ = 2w, and [GCH] is the statement that K’ = 2’( for all

The rôle of W. Wistar Comfort in the theory of topological groups

Abstract For a locally compact (LC) group G, denote by G+ its underlying group equipped with the topology inherited from its Bohr compactification. G is maximally almost periodic (MAP) if and only if G+ is Hausdorff. If P denotes a topological property, then we say that a MAP group G respects P if G and G+ have the same subspaces with P . In 1962 I. Glicksberg proved that LC Abelian groups respect compactness. We extend this result by showing that LC groups such that all their irreducible unitary representations are finite-dimensional, i.e., [MOORE] groups, do so as well. Moreover, we prove that G equipped with the topology induced by its topological dual is equal to G+ if and only if G belongs to the class [MOORE]. If this is indeed the case, then (a) G additionally respects pseudocompactness, (relative) functional boundedness, and the Lindelof property, (b) G is connected (respectively zero-dimensional, respectively realcompact) if and only if G+ is connected (respectively zero-dimensional, respectively realcompact), and (c) G is σ-compact if and only if G+ normal. We end the paper by showing the existence of a discrete group that is not [MOORE] and which still respects compactness.

Intervals of Totally Bounded Group Topologies

We prove that every countable free group admits at least ℵO non-isomorphic precompact totally minimal group topologies. Then we study the structure of the partially ordered set of T2- precompact group topologies on G∈C. C means the class of all infinite discrete maximally almost periodic groups G such that the quotient group with respect to the commutator subgroup has cardinality |G|. We apply the results to free groups. Finally, we determine the number of linear T2-precompact group topologies on free groups.

Extending Ring Topologies

Abstract We continue the work initiated in our earlier article (J. Pure Appl. Algebra 70 (1991) 53–72); as there, for G a group let B (G) (respectively N (G)) be the set of Hausdorff group topologies on G which are (respectively are not) totally bounded. In this abstract let A be the class of (discrete) maximally almost periodic groups G such that ¦G¦ = ¦ G G′ ¦ . We show (Theorem 3.3(A)) for G ϵ A with ¦G¦ = γ ⩾ ω that the condition that B (G) contains a chain C with ¦C¦ = β is equivalent to a natural and purely set-theoretic condition, namely that the partially ordered set 〈 P (2 γ ), ⊆ 〉 contains a chain of length β. (Thus the algebraic structure of G is irrelevant.) Similar results hold for chains in B (G) of fixed local weight, and for chains in N (G). Theorem 6.4. If T 1 ϵ B (G) and the Weil completion 〈(G,T 1 〉 is connected, then for every Hausdorff group topology T 0 ⊆ T 1 with ω〈G, T 0 〉 1 = ω〈G, T 1 〉 there are 2 α1 -many gro topologies between T 0 and T 1 . From Theorem 7.4. Let F be a compact, connected Lie group with trivial center. Then the product topology T 0 on F ω is the only pseudocompact group topology on F ω , but there are chains C ⊆ B (F ω ) and C ′ ⊆ B (F ω ) with ¦ C ¦ = (2 c+ and ¦ C ′¦ = 2 ( c+ ) such that T 0 ⊆ ∩ C and T 0 ⊆ ∩ C ′.

Abelian torsion groups with a pseudocompact group topology.

Abstract The important role of W.W. Comfort in the area of topological groups is described.