Inequalities in l₁ and l_p and applications to group algebras

Abstract

In this note, we show that, if (an) in l1 with Σ|an| < 2 and Σ|an|2 = 1, then max {|ai| + |aj|:i ≠ j} ≧ 1, but that the corresponding theorem for sequences in lp(1<p<2) fails—but only just! Applications to group algebras are given, when it is shown that elements in l1(G) with powers bounded by ½(1+ ) are bounded away from the identity e of G, but that the corresponding result for lp (G) is false.