Topological groups, limits and Pontryagin duality
by
Sergio Ardanza-Trevijano
Universidad de Navarra
Coauthors: María Jesús Chasco

Given a topological group G we denote by G^Ù its character group endowed with the compact open topology where by character we mean continuous homomorphisms from G into the multiplicative group T={z Î C : |z|=1} with the euclidean topology.

An abelian topological group is reflexive if the evaluation map, a_G : G® G^ÙÙ, defined by a_G(g)=á-, gñ, where ác, gñ = c(g) for each character c Î G^Ù is a topological isomorphism.

Pontryagin-van Kampen duality theorem states that locally compact abelian groups are reflexive. There have been different attempts to generalize this theorem to other situations. Kaplan proved that direct and inverse limits of sequences of locally compact abelian groups are reflexive. The argument in his proof relies deeply in the structure of the groups considered which are are locally compact. We will present some recent results showing that under certain conditions, sequences of non necessarily locally compact topological groups have reflexive limits and colimits.

Date received: June 14, 2004