Self-duality in the class of precompact groups

Abstract

A topological Abelian group G is called (strongly) self-dual if there exists a topological isomorphism Φ : G → G^∧ of G onto the dual group G^∧ (such that Φ (x) (y) = Φ (y) (x) for all x, y ∈ G). We prove that every countably compact self-dual Abelian group is finite. It turns out, however, that for every infinite cardinal κ with κ^ω = κ, there exists a pseudocompact, non-compact, strongly self-dual Boolean group of cardinality κ. © 2009 Elsevier B.V. All rights reserved.