A dichotomy property for locally compact groups.

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 { g n } n < ω in a locally compact group G , then either { g n } 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. [ABSTRACT FROM AUTHOR]