A dichotomy property for locally compact groups

Abstract

We extend to metrizable locally compact groups Rosenthal's theorem describing those Banach spaces containing no copy of ℓ1. For that purpose, we transfer to general locally compact groups the notion of interpolation (I0) set, which was defined by Hartman and Ryll-Nardzewsky [24] for locally compact abelian groups. Thus we prove that for every sequence {gn}n<ω in a locally compact group G, then either {gn}n<ω has a weak Cauchy subsequence or contains a subsequence that is an I0 set. This result is subsequently applied to obtain sufficient conditions for the existence of Sidon sets in a locally compact group G, an old question that remains open since 1974 (see [31] and [19]). Finally, we show that every locally compact group strongly respects compactness extending thereby a result by Comfort, Trigos-Arrieta, and Wu [13], who established this property for abelian locally compact groups.