A dichotomy property for locally compact groups

摘要

We extend to metrizable locally compact groups Rosenthal's theorem describing those Banach spaces containing no copy of l_1. For that purpose, we transfer to general locally compact groups the notion of interpolation (I_0) 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 I_0 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.